Areal or lattice data arise when a study region is divided into a finite number of areas where outcomes are aggregated. Examples include the number of individuals with a certain disease in municipalities of a country, the number of road accidents in provinces, or the average housing prices in districts of a city.
library(spData)Warning: package 'spData' was built under R version 4.4.2library(sf)Warning: package 'sf' was built under R version 4.4.3Linking to GEOS 3.13.0, GDAL 3.10.1, PROJ 9.5.1; sf_use_s2() is TRUElibrary(spdep)Warning: package 'spdep' was built under R version 4.4.2library(ggplot2)
map <- st_read(system.file("shapes/columbus.gpkg",
package = "spData"), quiet = TRUE)
st_crs(map) <- NANeighbors based on contiguity are constructed by assuming that neighbors of a given area are other areas that share a common boundary.
Here, we use poly2nb() to calculate the neighbors of each region in Columbus based on Queen contiguity.
library(spdep)
nb <- spdep::poly2nb(map, queen = TRUE)
head(nb)[[1]]
[1] 2 3
[[2]]
[1] 1 3 4
[[3]]
[1] 1 2 4 5
[[4]]
[1] 2 3 5 8
[[5]]
[1] 3 4 6 8 9 11 15 16
[[6]]
[1] 5 9plot(st_geometry(map), border = "lightgray")
plot.nb(nb, st_geometry(map), add = TRUE)
Map of neighbors based on contiguity.
We can plot the neighbors of a given area by adding a new column in map representing the neighbors of the area.
id <- 20 # area id
map$neighbors <- "other"
map$neighbors[id] <- "area"
map$neighbors[nb[[id]]] <- "neighbors"
ggplot(map) + geom_sf(aes(fill = neighbors)) +
theme_bw() + scale_fill_manual(values = c("gray30", "gray", "white"))
Map of neighbors of area 20 based on contiguity.
We can also consider as neighbors of an area its k nearest neighbors based on the distance separating them.
# Neighbors based on 3 nearest neighbors
coo <- st_centroid(map)Warning: st_centroid assumes attributes are constant over geometriesnb <- knn2nb(knearneigh(coo, k=3)) # k number nearest neighbors
plot(st_geometry(map), border = "lightgray")
plot.nb(nb, st_geometry(map), add = TRUE)
Map of neighbors based on 3 nearest neighbors
Neigborhood structures can also be defined by considering neighbors areas that are within a given distance.
# Neighbors based on distance
nb <- dnearneigh(x= st_centroid(map), d1 = 0, d2 = 0.4)Warning: st_centroid assumes attributes are constant over geometriesWarning in dnearneigh(x = st_centroid(map), d1 = 0, d2 = 0.4): neighbour object
has 18 sub-graphsplot(st_geometry(map), border = "lightgray")
plot.nb(nb, st_geometry(map), add = TRUE)
Map of neighbors separated by a distance less than 0.4.
Note that we can also determine an appropriate upper distance to ensure that each area has at least k neighbors. This helps in setting a suitable upper distance bound for a preferred number of neighbors k.
coo <- st_centroid(map)Warning: st_centroid assumes attributes are constant over geometries# k is the number nearest neighbors
nb1 <- knn2nb(knearneigh(coo, k=1))Warning in knn2nb(knearneigh(coo, k = 1)): neighbour object has 13 sub-graphsdist1 <- nbdists(nb1, coo)
summary(unlist(dist1)) Min. 1st Qu. Median Mean 3rd Qu. Max.
0.1276 0.2543 0.3164 0.3291 0.4044 0.6189 The maximum distance is 0.62 and we can take this value as an upper bound of the distance to ensure each area has at least one neighbor.
library(spdep)
nb <- poly2nb(map, queen = TRUE)
nblags <- spdep::nblag(neighbours = nb, maxlag = 2)# Neighbors of first order
plot(st_geometry(map), border = "lightgray")
plot.nb(nblags[[1]], st_geometry(map), add = TRUE)
Neighbors of first order.
# Neighbors of second order
plot(st_geometry(map), border = "lightgray")
plot.nb(nblags[[2]], st_geometry(map), add = TRUE)
Neighbors of second order.
# Neighbors of order 1 until order 2
nb <- spdep::poly2nb(map, queen = TRUE)
nblagsc <- spdep::nblag_cumul(nblags)
plot(st_geometry(map), border = "lightgray")
plot.nb(nblagsc, st_geometry(map), add = TRUE)
Neighbors of first order until second order.
A spatial neighborhood matrix (W) defines the neighborhood structure across the entire study region, where its elements represent spatial weights. The (i, j)-th element of W, denoted as w??????, expresses the spatial connection between areas i and j. Areas that are closer to i are assigned higher weights than those that are farther away.
Spatial weights matrix based on a binary neighbor list
nb <- poly2nb(map, queen = TRUE)
nbw <- spdep::nb2listw(nb, style = "W")
nbw$weights[1:3][[1]]
[1] 0.5 0.5
[[2]]
[1] 0.3333333 0.3333333 0.3333333
[[3]]
[1] 0.25 0.25 0.25 0.25m1 <- listw2mat(nbw)
lattice::levelplot(
t(m1),
scales = list(
y = list(
at = c(10, 20, 30, 40),
labels = c(10, 20, 30, 40)
)
)
)
Spatial weights matrix based on a binary neighbor list
Spatial weights matrix based on inverse distance values
coo <- st_centroid(map)Warning: st_centroid assumes attributes are constant over geometriesnb <- poly2nb(map, queen = TRUE)
dists <- nbdists(nb, coo)
ids <- lapply(dists, function(x){1/x})nbw <- nb2listw(nb, glist = ids, style = "B")
nbw$weights[1:3][[1]]
[1] 1.670235 1.725073
[[2]]
[1] 1.670235 1.404736 2.943309
[[3]]
[1] 1.725073 1.404736 1.782670 1.910694m2 <- listw2mat(nbw)
lattice::levelplot(
t(m2),
scales = list(
y = list(
at = c(10, 20, 30, 40),
labels = c(10, 20, 30, 40)
)
)
)
Spatial weights matrix based on inverse distance values
Spatial autocorrelation describes the extent to which a variable is correlated with itself across space. This concept is closely related to Tobler’s First Law of Geography, which states that “everything is related to everything else, but near things are more related than distant things.”
library(spData)
library(sf)
library(mapview)
map <- st_read(system.file("shapes/boston_tracts.gpkg", package = "spData"), quiet = TRUE)
map$vble <- map$MEDV
mapview(map, zcol = "vble")Median prices of owner-occupied housing in $1000 USD in census tracts of Boston in 1978.
where n is the number of regions, Y??? is the observed value of the variable of interest in region i, and ?? is the mean of all values. w?????? are the spatial weights that represent the spatial proximity between regions i and j, with w?????? = 0 and i, j = 1, ., n.
When the number of regions is sufficiently large, I follows a normal distribution. We can then assess whether a given spatial pattern significantly deviates from a random pattern by comparing its z-score.
# Neighbors
library(spdep)
nb <- poly2nb(map, queen = TRUE) # queen shares point or border
nbw <- nb2listw(nb, style = "W")
# Global Moran's I
gmoran <- moran.test(map$vble, nbw,
alternative = "greater")
gmoran
Moran I test under randomisation
data: map$vble
weights: nbw
Moran I statistic standard deviate = 23.35, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Moran I statistic Expectation Variance
0.6266753872 -0.0019801980 0.0007248686 gmoran[["estimate"]][["Moran I statistic"]] # Moran's I[1] 0.6266754gmoran[["statistic"]] # z-scoreMoran I statistic standard deviate
23.3498 gmoran[["p.value"]] # p-value[1] 6.923334e-121We observe the p-value obtained is lower than the significance level 0.05. Then, we reject the null hypothesis and conclude there is evidence for positive spatial autocorrelation.
The same conclusion is obtained if we use a Monte Carlo approach to assess significance.
gmoranMC <- moran.mc(map$vble, nbw, nsim = 999)
gmoranMC
Monte-Carlo simulation of Moran I
data: map$vble
weights: nbw
number of simulations + 1: 1000
statistic = 0.62668, observed rank = 1000, p-value = 0.001
alternative hypothesis: greaterhist(gmoranMC$res)
abline(v=gmoranMC$statistic, col = "red")
Histogram of the Moran’s I values for each of the simulated patterns in the Monte Carlo randomization approach. The red line represents the Moran’s I obtained for the real data.
This plot displays the observations of each area against its spatially lagged values. The spatially lagged value for a given area is calculated as a weighted average of the neighboring values for that area.
moran.plot(map$vble, nbw)
Moran’s I scatterplot showing the observations plotted against their spatially lagged values.
There is often interest in providing a local measure of similarity between each area’s value and those of its nearby areas. Local Indicators of Spatial Association (LISA) are designed to indicate the extent of significant spatial clustering of similar values around each observation. A desirable property of LISA is that the sum of all local indicators across regions equals a multiple of the global spatial association indicator.
Typically, the values of the LISAs are mapped to show the locations of areas with comparatively high or low local association with their neighboring areas.
Here, we use the localmoran() function to compute the Local Moran’s I for the housing prices data. We set alternative = "greater", which corresponds to testing H0: no or negative spatial autocorrelation vs. H1: positive spatial autocorrelation.
lmoran <- localmoran(map$vble, nbw, alternative = "greater")
head(lmoran) Ii E.Ii Var.Ii Z.Ii Pr(z > E(Ii))
1 -0.3457508492 -5.254157e-04 3.275376e-02 -1.90753363 9.717742e-01
2 0.0175875407 -1.626873e-05 2.045711e-03 0.38921049 3.485602e-01
3 0.0123379633 -6.557001e-07 4.089699e-05 1.92939381 2.684100e-02
4 -0.0001654033 -1.059064e-07 1.331742e-05 -0.04529559 5.180641e-01
5 0.3591628595 -1.427815e-04 7.898947e-03 4.04277384 2.641128e-05
6 0.0545610965 -1.625936e-04 1.357382e-02 0.46970410 3.192832e-01library(tmap)Warning: package 'tmap' was built under R version 4.4.2tmap_mode("plot")ℹ tmap mode set to "plot".map$lmI <- lmoran[, "Ii"] # Local Moran's I
map$lmZ <- lmoran[, "Z.Ii"] # z-scores
map$lmp <- lmoran[, "Pr(z > E(Ii))"] # p-values (alternative = "greater")
p1 <- tm_shape(map) +
tm_polygons(col = "vble", title = "vble", style = "quantile") +
tm_layout(legend.outside = TRUE)── tmap v3 code detected ───────────────────────────────────────────────────────[v3->v4] `tm_polygons()`: instead of `style = "quantile"`, use fill.scale =
`tm_scale_intervals()`.
ℹ Migrate the argument(s) 'style' to 'tm_scale_intervals(<HERE>)'
[v3->v4] `tm_polygons()`: use 'fill' for the fill color of polygons/symbols
(instead of 'col'), and 'col' for the outlines (instead of 'border.col').
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'p2 <- tm_shape(map) +
tm_polygons(col = "lmI", title = "Local Moran's I", style = "quantile") +
tm_layout(legend.outside = TRUE)[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'p3 <- tm_shape(map) +
tm_polygons(col = "lmZ", title = "Z-score", breaks = c(-Inf, 1.65, Inf)) +
tm_layout(legend.outside = TRUE)[v3->v4] `tm_tm_polygons()`: migrate the argument(s) related to the scale of
the visual variable `fill` namely 'breaks' to fill.scale = tm_scale(<HERE>).
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'p4 <- tm_shape(map) +
tm_polygons(col = "lmp", title = "p-value", breaks = c(-Inf, 0.05, Inf)) +
tm_layout(legend.outside = TRUE)[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'tmap_arrange(p1, p2, p3, p4)[scale] tm_polygons:() the data variable assigned to 'fill' contains positive and negative values, so midpoint is set to 0. Set 'midpoint = NA' in 'fill.scale = tm_scale_intervals(<HERE>)' to use all visual values (e.g. colors)
[scale] tm_polygons:() the data variable assigned to 'fill' contains positive and negative values, so midpoint is set to 0. Set 'midpoint = NA' in 'fill.scale = tm_scale_intervals(<HERE>)' to use all visual values (e.g. colors)
Boston housing prices, Local Moran’s I, z-scores, and p-values.
tm_shape(map) +
tm_polygons(
col = "lmZ",
title = "Local Moran's I",
style = "fixed",
breaks = c(-Inf, -1.96, 1.96, Inf),
labels = c("Negative SAC", "No SAC", "Positive SAC"),
palette = c("blue", "white", "red")
) +
tm_layout(legend.outside = TRUE)── tmap v3 code detected ───────────────────────────────────────────────────────[v3->v4] `tm_polygons()`: instead of `style = "fixed"`, use fill.scale =
`tm_scale_intervals()`.
ℹ Migrate the argument(s) 'style', 'breaks', 'palette' (rename to 'values'),
'labels' to 'tm_scale_intervals(<HERE>)'
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'
Multiple palettes called "blue" found: "kovesi.blue", "tableau.blue". The first one, "kovesi.blue", is returned.
Boston areas showing negative, no, and positive spatial auto correlation according to the local Moran’s I.
Local Moran’s I allows us to identify the following types of clusters: - High-High: areas with high values surrounded by neighbors with high values. - High-Low: areas with high values surrounded by neighbors with low values. - Low-High: areas with low values surrounded by neighbors with high values. - Low-Low: areas with low values surrounded by neighbors with low values.
lmoran <- localmoran(map$vble, nbw, alternative = "two.sided")
head(lmoran) Ii E.Ii Var.Ii Z.Ii Pr(z != E(Ii))
1 -0.3457508492 -5.254157e-04 3.275376e-02 -1.90753363 5.645152e-02
2 0.0175875407 -1.626873e-05 2.045711e-03 0.38921049 6.971204e-01
3 0.0123379633 -6.557001e-07 4.089699e-05 1.92939381 5.368199e-02
4 -0.0001654033 -1.059064e-07 1.331742e-05 -0.04529559 9.638717e-01
5 0.3591628595 -1.427815e-04 7.898947e-03 4.04277384 5.282257e-05
6 0.0545610965 -1.625936e-04 1.357382e-02 0.46970410 6.385664e-01map$lmp <- lmoran[,5] #p-values are in column 5
mp <- moran.plot(as.vector(scale(map$vble)), nbw)
Moran’s I scatterplot showing the scaled values plotted against their spatially lagged values.
head(mp) x wx is_inf labels dfb.1_ dfb.x dffit
1 -0.51459745 0.67055821 FALSE 1 0.090884905 -4.681542e-02 0.102233800
2 -0.09055093 -0.19384431 FALSE 2 -0.015161538 1.374250e-03 -0.015223692
3 0.01817895 0.67735383 FALSE 3 0.060002252 1.091857e-03 0.060012186
4 0.00730596 -0.02259476 FALSE 4 -0.004886055 -3.573265e-05 -0.004886186
5 0.26825767 1.33622668 TRUE 5 0.107457903 2.885493e-02 0.111264586
6 -0.28626471 -0.19021998 FALSE 6 -0.003358658 9.624167e-04 -0.003493827
cov.r cook.d hat
1 0.9900174 5.193220e-03 0.002500662
2 1.0055204 1.160840e-04 0.001992521
3 0.9987358 1.797813e-03 0.001976939
4 1.0059201 1.196085e-05 0.001976390
5 0.9831866 6.131141e-03 0.002118784
6 1.0061090 6.115479e-06 0.002138557We create a variable called quadrant to denote the type of cluster for each area, based on its value, its spatially lagged value, and the corresponding p-value. Specifically: - Areas with quadrant = 1 correspond to High-High clusters. - Areas with quadrant = 2 correspond to Low-Low clusters. - Areas with quadrant = 3 correspond to High-Low clusters. - Areas with quadrant = 4 correspond to Low-High clusters. - Areas with quadrant = 5 are non-significant.
map$quadrant <- NA
# High-High
map[(mp$x >= 0 & mp$wx >= 0) & (map$lmp <= 0.05), "quadrant"] <- 1
# Low-Low
map[(mp$x <= 0 & mp$wx <= 0) & (map$lmp <= 0.05), "quadrant"] <- 2
# High-Low
map[(mp$x >= 0 & mp$wx <= 0) & (map$lmp <= 0.05), "quadrant"] <- 3
# Low-High
map[(mp$x <= 0 & mp$wx >= 0) & (map$lmp <= 0.05), "quadrant"] <- 4
# Non-significant
map[(map$lmp > 0.05), "quadrant"] <- 5tm_shape(map) +
tm_fill(
col = "quadrant",
title = "",
breaks = c(1, 2, 3, 4, 5, 6),
palette = c("red", "blue", "lightpink", "skyblue2", "white"),
labels = c("High-High", "Low-Low", "High-Low",
"Low-High", "Non-significant")
) +
tm_legend(text.size = 1) +
tm_borders(alpha = 0.5) +
tm_layout(frame = FALSE, title = "Clusters") +
tm_layout(legend.outside = TRUE)── tmap v3 code detected ───────────────────────────────────────────────────────[v3->v4] `tm_tm_polygons()`: migrate the argument(s) related to the scale of
the visual variable `fill` namely 'breaks', 'palette' (rename to 'values'),
'labels' to fill.scale = tm_scale(<HERE>).
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'
[v3->v4] `tm_borders()`: use `fill_alpha` instead of `alpha`.
[v3->v4] `tm_layout()`: use `tm_title()` instead of `tm_layout(title = )`
[v3->v4] `tm_legend()`: use 'tm_legend()' inside a layer function, e.g.
'tm_polygons(..., fill.legend = tm_legend())'
High-high, low-low, high-low, and low-high clusters detected in the Boston housing prices data
Bayesian hierarchical models (Banerjee et al., 2004) can be used to analyze areal data that arise when an outcome variable is aggregated into areas that form a partition of the study region.
A commonly used spatial model is the Besag-York-Molli� (BYM) model.
The model includes a spatial random effect (u???) that accounts for the spatial dependence between outcomes, indicating that areas located close to each other may have similar values. An additional unstructured exchangeable component (v???) is included to model uncorrelated random noise.
Bayesian hierarchical models can be fitted using methods such as Integrated Nested Laplace Approximation (INLA) and Markov Chain Monte Carlo (MCMC). INLA provides a fast, approximate Bayesian inference approach for latent Gaussian models, including generalized linear mixed models and spatial or spatio-temporal models.
# Run R As Administrator
#install.packages("INLA",
# repos = c("https://cloud.r-project.org",
# INLA = "https://inla.r-inla-download.org/R/stable"),
# type = "binary")library(sf)
library(spData)
map <- st_read(system.file("shapes/boston_tracts.GPKG",
package = "spData"), quiet = TRUE)library(mapview)
map$vble <- log(map$MEDV)
mapview(map, zcol = "vble")Logarithm of housing prices in Boston per census tract from the spData package.
library(GGally)Registered S3 method overwritten by 'GGally':
method from
+.gg ggplot2ggpairs(data = map, columns = c("vble", "CRIM", "RM"))
The relationship between the outcome variable - the logarithm of housing price (VBLE) - and the covariates per capita crime rate (CRIM) and number of rooms (RM).
library(spdep)
library(INLA)Warning: package 'INLA' was built under R version 4.4.2Loading required package: MatrixThis is INLA_24.12.11 built 2024-12-11 19:58:26 UTC.
- See www.r-inla.org/contact-us for how to get help.
- List available models/likelihoods/etc with inla.list.models()
- Use inla.doc(<NAME>) to access documentationnb <-poly2nb(map)
head(nb)[[1]]
[1] 2 3 6 8 311 313 314 369
[[2]]
[1] 1 3 4 6
[[3]]
[1] 1 2 4 5 369 371 375 376
[[4]]
[1] 2 3 5 6
[[5]]
[1] 3 4 6 7 375 376 411 413 418
[[6]]
[1] 1 2 4 5 7 8nb2INLA("map.adj", nb)
g <- inla.read.graph(filename = "map.adj")map$re_u <- 1:nrow(map)
map$re_v <- 1:nrow(map)formula <- vble ~ CRIM + RM +
f(re_u, model = "besag", graph = g, scale.model = TRUE) +
f(re_v, model = "iid")formula <- vble ~ CRIM + RM + f(re_u, model = "bym2", graph = g)res <- inla(formula, family = "gaussian", data = map,
control.predictor = list(compute = TRUE),
control.compute = list(return.marginals.predictor = TRUE))res$summary.fixed mean sd 0.025quant 0.5quant 0.975quant
(Intercept) 1.426854150 0.088534457 1.25310149 1.42687863 1.600466729
CRIM -0.007842695 0.001292705 -0.01037973 -0.00784233 -0.005307747
RM 0.260317207 0.014043671 0.23277841 0.26031322 0.287878834
mode kld
(Intercept) 1.426878553 2.281143e-10
CRIM -0.007842332 2.381539e-10
RM 0.260313228 2.294824e-10We observe an intercept of ??0??? = 1.427 with a 95% credible interval of (1.253, 1.600). The coefficient for crime (CRIM) is ??1 = -0.008 with a 95% credible interval of (-0.010, -0.005), indicating that crime is significantly and negatively associated with housing prices. Meanwhile, the coefficient for number of rooms (RM) is ??2 = 0.260 with a 95% credible interval of (0.233, 0.288), suggesting that the number of rooms is significantly and positively associated with housing prices. Overall, these results imply that both crime rate and number of rooms play important roles in explaining the spatial variation of housing prices.
summary(res$summary.fitted.values) mean sd 0.025quant 0.5quant
Min. :1.610 Min. :0.009883 Min. :1.589 Min. :1.610
1st Qu.:2.835 1st Qu.:0.009911 1st Qu.:2.813 1st Qu.:2.835
Median :3.054 Median :0.009920 Median :3.032 Median :3.054
Mean :3.035 Mean :0.009931 Mean :3.013 Mean :3.035
3rd Qu.:3.219 3rd Qu.:0.009930 3rd Qu.:3.198 3rd Qu.:3.219
Max. :3.912 Max. :0.011217 Max. :3.891 Max. :3.912
0.975quant mode
Min. :1.633 Min. :1.610
1st Qu.:2.857 1st Qu.:2.835
Median :3.075 Median :3.054
Mean :3.056 Mean :3.035
3rd Qu.:3.241 3rd Qu.:3.219
Max. :3.934 Max. :3.912 We can create variables with the posterior mean (PM) and lower (LL) and upper (UL) limits of 95% credible intervals.
# Posterior mean and 95% CI
map$PM <- res$summary.fitted.values[, "mean"]
map$LL <- res$summary.fitted.values[, "0.025quant"]
map$UL <- res$summary.fitted.values[, "0.975quant"]# Common legend
at <- seq(
min(c(map$PM, map$LL, map$UL)),
max(c(map$PM, map$LL, map$UL)),
length.out = 8
)
# Popup table
popuptable <- leafpop::popupTable(
dplyr::mutate_if(map, is.numeric, round, digits = 2),
zcol = c("TOWN", "vble", "CRIM", "RM", "PM", "LL", "UL"),
row.numbers = FALSE,
feature.id = FALSE
)
# Map visualizations
m1 <- mapview(
map,
zcol = "PM",
map.types = "CartoDB.Positron",
at = at,
popup = popuptable
)
m2 <- mapview(
map,
zcol = "LL",
map.types = "CartoDB.Positron",
at = at,
popup = popuptable
)
m3 <- mapview(
map,
zcol = "UL",
map.types = "CartoDB.Positron",
at = at,
popup = popuptable
)library(leafsync)Warning: package 'leafsync' was built under R version 4.4.2m <- leafsync::sync(m1,m2,m3,ncol=3)
mPosterior mean of the logarithm of housing prices (left), along with the lower (center) and upper (right) limits of the 95% credible intervals.
# Transformation of the marginals using inla.tmarginal()
# Example: transformation for the first area
# inla.tmarginal(function(x) exp(x), res$marginals.fitted.values[[1]])
# Transform all marginals
marginals <- lapply(
res$marginals.fitted.values,
FUN = function(marg) {
inla.tmarginal(function(x) exp(x), marg)
}
)
# Obtain summaries of the transformed marginals using inla.zmarginal()
marginals_summaries <- lapply(
marginals,
FUN = function(marg) {
inla.zmarginal(marg)
}
)Mean 17.8214
Stdev 0.176978
Quantile 0.025 17.4708
Quantile 0.25 17.7288
Quantile 0.5 17.8116
Quantile 0.75 17.9004
Quantile 0.975 18.2401
Mean 21.7029
Stdev 0.213683
Quantile 0.025 21.2434
Quantile 0.25 21.597
Quantile 0.5 21.7005
Quantile 0.75 21.805
Quantile 0.975 22.1753
Mean 22.6976
Stdev 0.222831
Quantile 0.025 22.2102
Quantile 0.25 22.5884
Quantile 0.5 22.6974
Quantile 0.75 22.8058
Quantile 0.975 23.1818
Mean 22.5978
Stdev 0.222391
Quantile 0.025 22.1114
Quantile 0.25 22.489
Quantile 0.5 22.5975
Quantile 0.75 22.7056
Quantile 0.975 23.0814
Mean 25.0053
Stdev 0.245537
Quantile 0.025 24.4803
Quantile 0.25 24.8829
Quantile 0.5 25.0018
Quantile 0.75 25.1223
Quantile 0.975 25.5507
Mean 19.9089
Stdev 0.196105
Quantile 0.025 19.4969
Quantile 0.25 19.81
Quantile 0.5 19.9042
Quantile 0.75 20.0009
Quantile 0.975 20.3517
Mean 20.7922
Stdev 0.204449
Quantile 0.025 20.3364
Quantile 0.25 20.6936
Quantile 0.5 20.7943
Quantile 0.75 20.893
Quantile 0.975 21.2279
Mean 16.8141
Stdev 0.165825
Quantile 0.025 16.4762
Quantile 0.25 16.7287
Quantile 0.5 16.8074
Quantile 0.75 16.8901
Quantile 0.975 17.198
Mean 22.0037
Stdev 0.249736
Quantile 0.025 21.6288
Quantile 0.25 21.8587
Quantile 0.5 21.9585
Quantile 0.75 22.0884
Quantile 0.975 22.6811
Mean 27.4507
Stdev 0.274942
Quantile 0.025 26.7834
Quantile 0.25 27.3296
Quantile 0.5 27.4696
Quantile 0.75 27.5965
Quantile 0.975 27.9755
Mean 21.9174
Stdev 0.216892
Quantile 0.025 21.4736
Quantile 0.25 21.8062
Quantile 0.5 21.909
Quantile 0.75 22.0167
Quantile 0.975 22.4182
Mean 23.0788
Stdev 0.227902
Quantile 0.025 22.5524
Quantile 0.25 22.9725
Quantile 0.5 23.0863
Quantile 0.75 23.1946
Quantile 0.975 23.5449
Mean 49.864
Stdev 0.512281
Quantile 0.025 48.5665
Quantile 0.25 49.6523
Quantile 0.5 49.9176
Quantile 0.75 50.146
Quantile 0.975 50.7714
Mean 49.9376
Stdev 0.495119
Quantile 0.025 48.7715
Quantile 0.25 49.7115
Quantile 0.5 49.9607
Quantile 0.75 50.1935
Quantile 0.975 50.9248
Mean 49.9617
Stdev 0.492362
Quantile 0.025 48.8356
Quantile 0.25 49.7299
Quantile 0.5 49.9749
Quantile 0.75 50.21
Quantile 0.975 50.9807
Mean 49.9923
Stdev 0.49268
Quantile 0.025 48.9105
Quantile 0.25 49.7523
Quantile 0.5 49.9928
Quantile 0.75 50.2318
Quantile 0.975 51.0597
Mean 49.8138
Stdev 0.532235
Quantile 0.025 48.4166
Quantile 0.25 49.6093
Quantile 0.5 49.8887
Quantile 0.75 50.116
Quantile 0.975 50.6855
Mean 13.8033
Stdev 0.136253
Quantile 0.025 13.5125
Quantile 0.25 13.7355
Quantile 0.5 13.8012
Quantile 0.75 13.8679
Quantile 0.975 14.1068
Mean 13.8043
Stdev 0.136216
Quantile 0.025 13.5151
Quantile 0.25 13.7363
Quantile 0.5 13.8018
Quantile 0.75 13.8686
Quantile 0.975 14.1092
Mean 15.0158
Stdev 0.149123
Quantile 0.025 14.7169
Quantile 0.25 14.9384
Quantile 0.5 15.0085
Quantile 0.75 15.0829
Quantile 0.975 15.3657
Mean 13.8993
Stdev 0.136983
Quantile 0.025 13.6008
Quantile 0.25 13.8322
Quantile 0.5 13.8989
Quantile 0.75 13.9655
Quantile 0.975 14.1984
Mean 13.3036
Stdev 0.130981
Quantile 0.025 13.0248
Quantile 0.25 13.2382
Quantile 0.5 13.3014
Quantile 0.75 13.3657
Quantile 0.975 13.5959
Mean 13.1065
Stdev 0.129069
Quantile 0.025 12.8363
Quantile 0.25 13.0412
Quantile 0.5 13.1031
Quantile 0.75 13.1668
Quantile 0.975 13.3988
Mean 10.2029
Stdev 0.100589
Quantile 0.025 9.98885
Quantile 0.25 10.1527
Quantile 0.5 10.2011
Quantile 0.75 10.2505
Quantile 0.975 10.4275
Mean 10.3978
Stdev 0.102377
Quantile 0.025 10.172
Quantile 0.25 10.348
Quantile 0.5 10.3981
Quantile 0.75 10.4477
Quantile 0.975 10.6186
Mean 10.9144
Stdev 0.108686
Quantile 0.025 10.7012
Quantile 0.25 10.8573
Quantile 0.5 10.9079
Quantile 0.75 10.9625
Quantile 0.975 11.1734
Mean 11.2989
Stdev 0.111196
Quantile 0.025 11.0558
Quantile 0.25 11.2445
Quantile 0.5 11.2988
Quantile 0.75 11.3528
Quantile 0.975 11.5408
Mean 12.2996
Stdev 0.120901
Quantile 0.025 12.0366
Quantile 0.25 12.2402
Quantile 0.5 12.2991
Quantile 0.75 12.3581
Quantile 0.975 12.5638
Mean 8.79668
Stdev 0.0865491
Quantile 0.025 8.60373
Quantile 0.25 8.755
Quantile 0.5 8.79758
Quantile 0.75 8.83936
Quantile 0.975 8.98112
Mean 7.20689
Stdev 0.0715166
Quantile 0.025 7.06239
Quantile 0.25 7.16996
Quantile 0.5 7.20365
Quantile 0.75 7.23927
Quantile 0.975 7.3737
Mean 10.4887
Stdev 0.103691
Quantile 0.025 10.247
Quantile 0.25 10.4408
Quantile 0.5 10.4928
Quantile 0.75 10.5419
Quantile 0.975 10.6982
Mean 7.40513
Stdev 0.0732461
Quantile 0.025 7.25406
Quantile 0.25 7.36779
Quantile 0.5 7.40261
Quantile 0.75 7.43888
Quantile 0.975 7.5732
Mean 10.1984
Stdev 0.1003
Quantile 0.025 9.97809
Quantile 0.25 10.1495
Quantile 0.5 10.1985
Quantile 0.75 10.2472
Quantile 0.975 10.4155
Mean 11.4992
Stdev 0.112892
Quantile 0.025 11.2529
Quantile 0.25 11.4438
Quantile 0.5 11.4989
Quantile 0.75 11.5539
Quantile 0.975 11.7452
Mean 15.1009
Stdev 0.148368
Quantile 0.025 14.7801
Quantile 0.25 15.0276
Quantile 0.5 15.0997
Quantile 0.75 15.1722
Quantile 0.975 15.4271
Mean 23.1715
Stdev 0.229407
Quantile 0.025 22.6318
Quantile 0.25 23.0665
Quantile 0.5 23.182
Quantile 0.75 23.29
Quantile 0.975 23.6294
Mean 9.7048
Stdev 0.0954115
Quantile 0.025 9.50508
Quantile 0.25 9.65651
Quantile 0.5 9.70231
Quantile 0.75 9.74949
Quantile 0.975 9.92081
Mean 13.7925
Stdev 0.135886
Quantile 0.025 13.486
Quantile 0.25 13.7277
Quantile 0.5 13.7948
Quantile 0.75 13.8601
Quantile 0.975 14.0785
Mean 12.6955
Stdev 0.125064
Quantile 0.025 12.4171
Quantile 0.25 12.6352
Quantile 0.5 12.6967
Quantile 0.75 12.757
Quantile 0.975 12.9625
Mean 13.0993
Stdev 0.129006
Quantile 0.025 12.8181
Quantile 0.25 13.0361
Quantile 0.5 13.0989
Quantile 0.75 13.1617
Quantile 0.975 13.3808
Mean 12.4957
Stdev 0.123008
Quantile 0.025 12.2222
Quantile 0.25 12.4364
Quantile 0.5 12.4968
Quantile 0.75 12.5562
Quantile 0.975 12.7586
Mean 8.50453
Stdev 0.0838291
Quantile 0.025 8.32954
Quantile 0.25 8.46208
Quantile 0.5 8.5022
Quantile 0.75 8.54362
Quantile 0.975 8.69486
Mean 5.00484
Stdev 0.0495816
Quantile 0.025 4.90478
Quantile 0.25 4.97919
Quantile 0.5 5.00257
Quantile 0.75 5.0273
Quantile 0.975 5.12055
Mean 6.30237
Stdev 0.0621336
Quantile 0.025 6.17113
Quantile 0.25 6.27119
Quantile 0.5 6.30105
Quantile 0.75 6.33161
Quantile 0.975 6.44204
Mean 5.60803
Stdev 0.0559972
Quantile 0.025 5.49912
Quantile 0.25 5.57844
Quantile 0.5 5.6044
Quantile 0.75 5.63258
Quantile 0.975 5.74226
Mean 7.20592
Stdev 0.0712434
Quantile 0.025 7.06049
Quantile 0.25 7.16932
Quantile 0.5 7.20309
Quantile 0.75 7.23852
Quantile 0.975 7.3707
Mean 12.0942
Stdev 0.119042
Quantile 0.025 11.8269
Quantile 0.25 12.0372
Quantile 0.5 12.0959
Quantile 0.75 12.1532
Quantile 0.975 12.3459
Mean 8.29911
Stdev 0.0816742
Quantile 0.025 8.1204
Quantile 0.25 8.25918
Quantile 0.5 8.29904
Quantile 0.75 8.33873
Quantile 0.975 8.47663
Mean 8.49486
Stdev 0.083617
Quantile 0.025 8.30561
Quantile 0.25 8.45511
Quantile 0.5 8.49653
Quantile 0.75 8.53665
Quantile 0.975 8.67005
Mean 5.00366
Stdev 0.0494342
Quantile 0.025 4.90201
Quantile 0.25 4.97838
Quantile 0.5 5.00188
Quantile 0.75 5.02641
Quantile 0.975 5.11734
Mean 11.9039
Stdev 0.117557
Quantile 0.025 11.6545
Quantile 0.25 11.8451
Quantile 0.5 11.9016
Quantile 0.75 11.9593
Quantile 0.975 12.1673
Mean 27.8914
Stdev 0.27403
Quantile 0.025 27.2834
Quantile 0.25 27.7588
Quantile 0.5 27.8935
Quantile 0.75 28.0261
Quantile 0.975 28.4782
Mean 17.213
Stdev 0.170151
Quantile 0.025 16.8638
Quantile 0.25 17.1259
Quantile 0.5 17.2067
Quantile 0.75 17.2911
Quantile 0.975 17.6049
Mean 27.4996
Stdev 0.27049
Quantile 0.025 26.9116
Quantile 0.25 27.3666
Quantile 0.5 27.4983
Quantile 0.75 27.6301
Quantile 0.975 28.0913
Mean 15.0071
Stdev 0.148068
Quantile 0.025 14.6967
Quantile 0.25 14.9325
Quantile 0.5 15.0034
Quantile 0.75 15.0764
Quantile 0.975 15.3421
Mean 17.235
Stdev 0.174531
Quantile 0.025 16.9119
Quantile 0.25 17.1405
Quantile 0.5 17.2196
Quantile 0.75 17.3083
Quantile 0.975 17.6666
Mean 17.8788
Stdev 0.177112
Quantile 0.025 17.4632
Quantile 0.25 17.7976
Quantile 0.5 17.8866
Quantile 0.75 17.9701
Quantile 0.975 18.2337
Mean 16.2783
Stdev 0.161604
Quantile 0.025 15.8958
Quantile 0.25 16.2049
Quantile 0.5 16.2864
Quantile 0.75 16.3622
Quantile 0.975 16.5984
Mean 6.99166
Stdev 0.0691612
Quantile 0.025 6.82933
Quantile 0.25 6.95993
Quantile 0.5 6.99472
Quantile 0.75 7.02735
Quantile 0.975 7.13014
Mean 7.20559
Stdev 0.0712911
Quantile 0.025 7.05952
Quantile 0.25 7.16908
Quantile 0.5 7.20289
Quantile 0.75 7.23828
Quantile 0.975 7.37003
Mean 7.51283
Stdev 0.0755261
Quantile 0.025 7.3692
Quantile 0.25 7.47248
Quantile 0.5 7.50711
Quantile 0.75 7.54528
Quantile 0.975 7.6966
Mean 10.4088
Stdev 0.103071
Quantile 0.025 10.1988
Quantile 0.25 10.3558
Quantile 0.5 10.4046
Quantile 0.75 10.4558
Quantile 0.975 10.6476
Mean 8.79058
Stdev 0.0869942
Quantile 0.025 8.58782
Quantile 0.25 8.75041
Quantile 0.5 8.79399
Quantile 0.75 8.83513
Quantile 0.975 8.96648
Mean 8.40673
Stdev 0.0833143
Quantile 0.025 8.23633
Quantile 0.25 8.36404
Quantile 0.5 8.40349
Quantile 0.75 8.44483
Quantile 0.975 8.59924
Mean 16.6866
Stdev 0.164805
Quantile 0.025 16.3087
Quantile 0.25 16.6093
Quantile 0.5 16.6912
Quantile 0.75 16.7697
Quantile 0.975 17.0268
Mean 14.2163
Stdev 0.141215
Quantile 0.025 13.9353
Quantile 0.25 14.1426
Quantile 0.5 14.2088
Quantile 0.75 14.2795
Quantile 0.975 14.5494
Mean 20.7735
Stdev 0.205927
Quantile 0.025 20.2877
Quantile 0.25 20.6795
Quantile 0.5 20.7833
Quantile 0.75 20.8801
Quantile 0.975 21.1831
Mean 13.4091
Stdev 0.13236
Quantile 0.025 13.136
Quantile 0.25 13.3416
Quantile 0.5 13.4046
Quantile 0.75 13.4703
Quantile 0.975 13.7126
Mean 11.7001
Stdev 0.114943
Quantile 0.025 11.4507
Quantile 0.25 11.6435
Quantile 0.5 11.6994
Quantile 0.75 11.7555
Quantile 0.975 11.9519
Mean 8.31729
Stdev 0.0842312
Quantile 0.025 8.16189
Quantile 0.25 8.27157
Quantile 0.5 8.30968
Quantile 0.75 8.35256
Quantile 0.975 8.52596
Mean 10.2027
Stdev 0.100448
Quantile 0.025 9.98886
Quantile 0.25 10.1526
Quantile 0.5 10.2011
Quantile 0.75 10.2504
Quantile 0.975 10.4269
Mean 10.8964
Stdev 0.107263
Quantile 0.025 10.6579
Quantile 0.25 10.8446
Quantile 0.5 10.8973
Quantile 0.75 10.9491
Quantile 0.975 11.1257
Mean 11.002
Stdev 0.108215
Quantile 0.025 10.7702
Quantile 0.25 10.9482
Quantile 0.5 11.0006
Quantile 0.75 11.0536
Quantile 0.975 11.2421
Mean 9.50502
Stdev 0.0935241
Quantile 0.025 9.30978
Quantile 0.25 9.45762
Quantile 0.5 9.50244
Quantile 0.75 9.5487
Quantile 0.975 9.71728
Mean 14.4952
Stdev 0.14242
Quantile 0.025 14.1786
Quantile 0.25 14.4263
Quantile 0.5 14.4964
Quantile 0.75 14.5653
Quantile 0.975 14.7996
Mean 14.0993
Stdev 0.138584
Quantile 0.025 13.7973
Quantile 0.25 14.0313
Quantile 0.5 14.0988
Quantile 0.75 14.1664
Quantile 0.975 14.4017
Mean 16.0968
Stdev 0.158313
Quantile 0.025 15.7483
Quantile 0.25 16.0198
Quantile 0.5 16.0973
Quantile 0.75 16.1741
Quantile 0.975 16.4387
Mean 14.303
Stdev 0.140687
Quantile 0.025 14.0022
Quantile 0.25 14.233
Quantile 0.5 14.301
Quantile 0.75 14.37
Quantile 0.975 14.6156
Mean 11.7023
Stdev 0.115098
Quantile 0.025 11.4558
Quantile 0.25 11.645
Quantile 0.5 11.7007
Quantile 0.75 11.7571
Quantile 0.975 11.9577
Mean 13.3967
Stdev 0.131604
Quantile 0.025 13.106
Quantile 0.25 13.3328
Quantile 0.5 13.3974
Quantile 0.75 13.4612
Quantile 0.975 13.6799
Mean 9.60468
Stdev 0.0946953
Quantile 0.025 9.40632
Quantile 0.25 9.55686
Quantile 0.5 9.60223
Quantile 0.75 9.64895
Quantile 0.975 9.81907
Mean 8.70338
Stdev 0.0857473
Quantile 0.025 8.52243
Quantile 0.25 8.6603
Quantile 0.5 8.70152
Quantile 0.75 8.74373
Quantile 0.975 8.89626
Mean 8.40506
Stdev 0.0828987
Quantile 0.025 8.23291
Quantile 0.25 8.36293
Quantile 0.5 8.40252
Quantile 0.75 8.44354
Quantile 0.975 8.59412
Mean 12.7919
Stdev 0.125794
Quantile 0.025 12.5066
Quantile 0.25 12.7321
Quantile 0.5 12.7945
Quantile 0.75 12.8549
Quantile 0.975 13.0548
Mean 10.5052
Stdev 0.103718
Quantile 0.025 10.288
Quantile 0.25 10.4528
Quantile 0.5 10.5025
Quantile 0.75 10.5536
Quantile 0.975 10.7402
Mean 17.0954
Stdev 0.167945
Quantile 0.025 16.7239
Quantile 0.25 17.014
Quantile 0.5 17.0964
Quantile 0.75 17.1778
Quantile 0.975 17.4562
Mean 18.3883
Stdev 0.181146
Quantile 0.025 17.9774
Quantile 0.25 18.3024
Quantile 0.5 18.3921
Quantile 0.75 18.4789
Quantile 0.975 18.7669
Mean 15.3995
Stdev 0.151368
Quantile 0.025 15.0702
Quantile 0.25 15.3252
Quantile 0.5 15.3989
Quantile 0.75 15.4727
Quantile 0.975 15.7303
Mean 10.7998
Stdev 0.106157
Quantile 0.025 10.569
Quantile 0.25 10.7476
Quantile 0.5 10.7993
Quantile 0.75 10.8511
Quantile 0.975 11.032
Mean 11.8034
Stdev 0.116075
Quantile 0.025 11.5566
Quantile 0.25 11.7453
Quantile 0.5 11.8014
Quantile 0.75 11.8584
Quantile 0.975 12.0627
Mean 14.8985
Stdev 0.146346
Quantile 0.025 14.5786
Quantile 0.25 14.8269
Quantile 0.5 14.8983
Quantile 0.75 14.9696
Quantile 0.975 15.2168
Mean 12.6062
Stdev 0.124211
Quantile 0.025 12.346
Quantile 0.25 12.5434
Quantile 0.5 12.6029
Quantile 0.75 12.6643
Quantile 0.975 12.8874
Mean 14.0958
Stdev 0.138745
Quantile 0.025 13.788
Quantile 0.25 14.0287
Quantile 0.5 14.0967
Quantile 0.75 14.1638
Quantile 0.975 14.3931
Mean 13.0024
Stdev 0.127801
Quantile 0.025 12.7285
Quantile 0.25 12.9388
Quantile 0.5 13.0007
Quantile 0.75 13.0633
Quantile 0.975 13.2857
Mean 13.4079
Stdev 0.132142
Quantile 0.025 13.1331
Quantile 0.25 13.3407
Quantile 0.5 13.4039
Quantile 0.75 13.4693
Quantile 0.975 13.7089
Mean 15.2012
Stdev 0.149452
Quantile 0.025 14.8786
Quantile 0.25 15.1273
Quantile 0.5 15.1999
Quantile 0.75 15.2729
Quantile 0.975 15.5303
Mean 16.0996
Stdev 0.158155
Quantile 0.025 15.7557
Quantile 0.25 16.0218
Quantile 0.5 16.0989
Quantile 0.75 16.1761
Quantile 0.975 16.4454
Mean 17.804
Stdev 0.175359
Quantile 0.025 17.4293
Quantile 0.25 17.7167
Quantile 0.5 17.8014
Quantile 0.75 17.8873
Quantile 0.975 18.1941
Mean 14.9005
Stdev 0.146392
Quantile 0.025 14.5835
Quantile 0.25 14.8283
Quantile 0.5 14.8995
Quantile 0.75 14.971
Quantile 0.975 15.2219
Mean 14.106
Stdev 0.138922
Quantile 0.025 13.8136
Quantile 0.25 14.036
Quantile 0.5 14.1028
Quantile 0.75 14.1712
Quantile 0.975 14.4192
Mean 12.7089
Stdev 0.125393
Quantile 0.025 12.4504
Quantile 0.25 12.6448
Quantile 0.5 12.7045
Quantile 0.75 12.7667
Quantile 0.975 12.9966
Mean 13.5061
Stdev 0.133214
Quantile 0.025 13.2264
Quantile 0.25 13.439
Quantile 0.5 13.5029
Quantile 0.75 13.5685
Quantile 0.975 13.8071
Mean 14.9126
Stdev 0.147393
Quantile 0.025 14.6123
Quantile 0.25 14.8368
Quantile 0.5 14.9066
Quantile 0.75 14.98
Quantile 0.975 15.254
Mean 19.9912
Stdev 0.196857
Quantile 0.025 19.5504
Quantile 0.25 19.8968
Quantile 0.5 19.9937
Quantile 0.75 20.0886
Quantile 0.975 20.4088
Mean 16.4073
Stdev 0.161332
Quantile 0.025 16.0684
Quantile 0.25 16.3259
Quantile 0.5 16.4034
Quantile 0.75 16.4831
Quantile 0.975 16.7715
Mean 17.7066
Stdev 0.174317
Quantile 0.025 17.3382
Quantile 0.25 17.619
Quantile 0.5 17.7029
Quantile 0.75 17.7887
Quantile 0.975 18.0981
Mean 19.5023
Stdev 0.192005
Quantile 0.025 19.0889
Quantile 0.25 19.4073
Quantile 0.5 19.5003
Quantile 0.75 19.5942
Quantile 0.975 19.9263
Mean 20.2049
Stdev 0.198651
Quantile 0.025 19.7812
Quantile 0.25 20.1058
Quantile 0.5 20.2018
Quantile 0.75 20.2994
Quantile 0.975 20.6473
Mean 21.3947
Stdev 0.2103
Quantile 0.025 20.9301
Quantile 0.25 21.2926
Quantile 0.5 21.3958
Quantile 0.75 21.4977
Quantile 0.975 21.8472
Mean 19.8911
Stdev 0.195292
Quantile 0.025 19.4538
Quantile 0.25 19.7973
Quantile 0.5 19.8937
Quantile 0.75 19.9879
Quantile 0.975 20.305
Mean 18.9913
Stdev 0.186788
Quantile 0.025 18.5726
Quantile 0.25 18.9017
Quantile 0.5 18.9939
Quantile 0.75 19.0839
Quantile 0.975 19.3869
Mean 19.08
Stdev 0.187685
Quantile 0.025 18.6435
Quantile 0.25 18.9928
Quantile 0.5 19.0872
Quantile 0.75 19.1763
Quantile 0.975 19.46
Mean 19.0971
Stdev 0.187705
Quantile 0.025 18.6851
Quantile 0.25 19.0055
Quantile 0.5 19.0973
Quantile 0.75 19.1885
Quantile 0.975 19.5037
Mean 20.0942
Stdev 0.197916
Quantile 0.025 19.6555
Quantile 0.25 19.9985
Quantile 0.5 20.0955
Quantile 0.75 20.1912
Quantile 0.975 20.5188
Mean 19.9072
Stdev 0.195852
Quantile 0.025 19.4931
Quantile 0.25 19.8089
Quantile 0.5 19.9032
Quantile 0.75 19.9996
Quantile 0.975 20.3468
Mean 19.6052
Stdev 0.193139
Quantile 0.025 19.1939
Quantile 0.25 19.5089
Quantile 0.5 19.602
Quantile 0.75 19.6968
Quantile 0.975 20.0361
Mean 23.2027
Stdev 0.227873
Quantile 0.025 22.7121
Quantile 0.25 23.0897
Quantile 0.5 23.2003
Quantile 0.75 23.3119
Quantile 0.975 23.7057
Mean 29.7858
Stdev 0.292635
Quantile 0.025 29.1291
Quantile 0.25 29.6455
Quantile 0.5 29.7901
Quantile 0.75 29.9311
Quantile 0.975 30.4047
Mean 13.8018
Stdev 0.13589
Quantile 0.025 13.5095
Quantile 0.25 13.7345
Quantile 0.5 13.8003
Quantile 0.75 13.8668
Quantile 0.975 14.1022
Mean 13.309
Stdev 0.131443
Quantile 0.025 13.0376
Quantile 0.25 13.242
Quantile 0.5 13.3046
Quantile 0.75 13.3697
Quantile 0.975 13.6102
Mean 16.7097
Stdev 0.164775
Quantile 0.025 16.367
Quantile 0.25 16.6261
Quantile 0.5 16.7048
Quantile 0.75 16.7863
Quantile 0.975 17.085
Mean 11.9964
Stdev 0.118007
Quantile 0.025 11.7347
Quantile 0.25 11.9393
Quantile 0.5 11.9972
Quantile 0.75 12.0543
Quantile 0.975 12.2493
Mean 14.6046
Stdev 0.143916
Quantile 0.025 14.2993
Quantile 0.25 14.5326
Quantile 0.5 14.6019
Quantile 0.75 14.6726
Quantile 0.975 14.9268
Mean 21.3866
Stdev 0.210538
Quantile 0.025 20.9094
Quantile 0.25 21.2866
Quantile 0.5 21.391
Quantile 0.75 21.4919
Quantile 0.975 21.8269
Mean 23.0107
Stdev 0.22642
Quantile 0.025 22.5356
Quantile 0.25 22.8963
Quantile 0.5 23.005
Quantile 0.75 23.1168
Quantile 0.975 23.5223
Mean 23.7056
Stdev 0.2335
Quantile 0.025 23.2072
Quantile 0.25 23.5893
Quantile 0.5 23.702
Quantile 0.75 23.8165
Quantile 0.975 24.2255
Mean 25.0065
Stdev 0.246185
Quantile 0.025 24.482
Quantile 0.25 24.8837
Quantile 0.5 25.0025
Quantile 0.75 25.1233
Quantile 0.975 25.5555
Mean 21.7935
Stdev 0.214375
Quantile 0.025 21.318
Quantile 0.25 21.6897
Quantile 0.5 21.795
Quantile 0.75 21.8987
Quantile 0.975 22.2529
Mean 20.5965
Stdev 0.202088
Quantile 0.025 20.1525
Quantile 0.25 20.4978
Quantile 0.5 20.5968
Quantile 0.75 20.6951
Quantile 0.975 21.0337
Mean 21.2132
Stdev 0.209139
Quantile 0.025 20.7796
Quantile 0.25 21.1068
Quantile 0.5 21.2066
Quantile 0.75 21.3102
Quantile 0.975 21.6907
Mean 19.1072
Stdev 0.188351
Quantile 0.025 18.7092
Quantile 0.25 19.0126
Quantile 0.5 19.1032
Quantile 0.75 19.1958
Quantile 0.975 19.5304
Mean 20.5989
Stdev 0.202468
Quantile 0.025 20.1575
Quantile 0.25 20.4995
Quantile 0.5 20.5982
Quantile 0.75 20.6969
Quantile 0.975 21.0405
Mean 15.1848
Stdev 0.150076
Quantile 0.025 14.8364
Quantile 0.25 15.1151
Quantile 0.5 15.1902
Quantile 0.75 15.2614
Quantile 0.975 15.4898
Mean 7.00998
Stdev 0.0700733
Quantile 0.025 6.87361
Quantile 0.25 6.973
Quantile 0.5 7.00547
Quantile 0.75 7.0407
Quantile 0.975 7.17792
Mean 8.11337
Stdev 0.0813952
Quantile 0.025 7.95784
Quantile 0.25 8.06997
Quantile 0.5 8.1074
Quantile 0.75 8.14852
Quantile 0.975 8.31079
Mean 13.6141
Stdev 0.134906
Quantile 0.025 13.3432
Quantile 0.25 13.544
Quantile 0.5 13.6075
Quantile 0.75 13.6749
Quantile 0.975 13.9301
Mean 20.091
Stdev 0.197477
Quantile 0.025 19.6487
Quantile 0.25 19.9962
Quantile 0.5 20.0936
Quantile 0.75 20.1888
Quantile 0.975 20.5096
Mean 21.7901
Stdev 0.214445
Quantile 0.025 21.3095
Quantile 0.25 21.6873
Quantile 0.5 21.793
Quantile 0.75 21.8963
Quantile 0.975 22.2445
Mean 24.494
Stdev 0.240914
Quantile 0.025 23.9617
Quantile 0.25 24.3771
Quantile 0.5 24.4952
Quantile 0.75 24.6119
Quantile 0.975 25.0124
Mean 23.0837
Stdev 0.227101
Quantile 0.025 22.5663
Quantile 0.25 22.9763
Quantile 0.5 23.0892
Quantile 0.75 23.1979
Quantile 0.975 23.5557
Mean 19.6949
Stdev 0.193596
Quantile 0.025 19.2669
Quantile 0.25 19.601
Quantile 0.5 19.696
Quantile 0.75 19.7898
Quantile 0.975 20.1111
Mean 18.3
Stdev 0.179996
Quantile 0.025 17.909
Quantile 0.25 18.2114
Quantile 0.5 18.299
Quantile 0.75 18.3867
Quantile 0.975 18.6941
Mean 21.1967
Stdev 0.207975
Quantile 0.025 20.7402
Quantile 0.25 21.0951
Quantile 0.5 21.1969
Quantile 0.75 21.2981
Quantile 0.975 21.647
Mean 17.5019
Stdev 0.172085
Quantile 0.025 17.1311
Quantile 0.25 17.4167
Quantile 0.5 17.5002
Quantile 0.75 17.5843
Quantile 0.975 17.8815
Mean 16.8077
Stdev 0.165895
Quantile 0.025 16.4594
Quantile 0.25 16.7241
Quantile 0.5 16.8036
Quantile 0.75 16.8853
Quantile 0.975 17.1827
Mean 22.3948
Stdev 0.220264
Quantile 0.025 21.9086
Quantile 0.25 22.2878
Quantile 0.5 22.3957
Quantile 0.75 22.5025
Quantile 0.975 22.8692
Mean 20.5996
Stdev 0.202744
Quantile 0.025 20.1587
Quantile 0.25 20.5
Quantile 0.5 20.5987
Quantile 0.75 20.6974
Quantile 0.975 21.043
Mean 23.894
Stdev 0.235019
Quantile 0.025 23.3746
Quantile 0.25 23.78
Quantile 0.5 23.8952
Quantile 0.75 24.009
Quantile 0.975 24.3996
Mean 21.9984
Stdev 0.216497
Quantile 0.025 21.5257
Quantile 0.25 21.8923
Quantile 0.5 21.9979
Quantile 0.75 22.1032
Quantile 0.975 22.47
Mean 11.9113
Stdev 0.118183
Quantile 0.025 11.6723
Quantile 0.25 11.8503
Quantile 0.5 11.906
Quantile 0.75 11.9648
Quantile 0.975 12.1868
Mean 23.9864
Stdev 0.236501
Quantile 0.025 23.4524
Quantile 0.25 23.8739
Quantile 0.5 23.9907
Quantile 0.75 24.1042
Quantile 0.975 24.4835
Mean 21.6058
Stdev 0.212328
Quantile 0.025 21.1538
Quantile 0.25 21.4997
Quantile 0.5 21.6023
Quantile 0.75 21.7066
Quantile 0.975 22.0795
Mean 34.6834
Stdev 0.341592
Quantile 0.025 33.9165
Quantile 0.25 34.5199
Quantile 0.5 34.6884
Quantile 0.75 34.8527
Quantile 0.975 35.4059
Mean 33.393
Stdev 0.328641
Quantile 0.025 32.6688
Quantile 0.25 33.2333
Quantile 0.5 33.3941
Quantile 0.75 33.5534
Quantile 0.975 34.1021
Mean 36.1867
Stdev 0.35625
Quantile 0.025 35.393
Quantile 0.25 36.0151
Quantile 0.5 36.1903
Quantile 0.75 36.3622
Quantile 0.975 36.9466
Mean 28.69
Stdev 0.282619
Quantile 0.025 28.0611
Quantile 0.25 28.5538
Quantile 0.5 28.6926
Quantile 0.75 28.829
Quantile 0.975 29.2937
Mean 22.9018
Stdev 0.225026
Quantile 0.025 22.416
Quantile 0.25 22.7905
Quantile 0.5 22.8998
Quantile 0.75 23.0099
Quantile 0.975 23.3972
Mean 27.0795
Stdev 0.267164
Quantile 0.025 26.4688
Quantile 0.25 26.9538
Quantile 0.5 27.0865
Quantile 0.75 27.214
Quantile 0.975 27.633
Mean 16.5119
Stdev 0.163053
Quantile 0.025 16.1763
Quantile 0.25 16.4285
Quantile 0.5 16.5061
Quantile 0.75 16.587
Quantile 0.975 16.8865
Mean 18.8997
Stdev 0.185894
Quantile 0.025 18.4957
Quantile 0.25 18.8083
Quantile 0.5 18.8988
Quantile 0.75 18.9894
Quantile 0.975 19.3065
Mean 15.022
Stdev 0.150088
Quantile 0.025 14.7308
Quantile 0.25 14.9426
Quantile 0.5 15.0121
Quantile 0.75 15.0876
Quantile 0.975 15.3823
Mean 18.8995
Stdev 0.185659
Quantile 0.025 18.4956
Quantile 0.25 18.8082
Quantile 0.5 18.8987
Quantile 0.75 18.9892
Quantile 0.975 19.3052
Mean 21.6837
Stdev 0.213172
Quantile 0.025 21.1967
Quantile 0.25 21.583
Quantile 0.5 21.6892
Quantile 0.75 21.7912
Quantile 0.975 22.1252
Mean 20.3935
Stdev 0.200366
Quantile 0.025 19.9485
Quantile 0.25 20.2966
Quantile 0.5 20.3951
Quantile 0.75 20.492
Quantile 0.975 20.8222
Mean 18.2015
Stdev 0.179062
Quantile 0.025 17.8149
Quantile 0.25 18.113
Quantile 0.5 18.1999
Quantile 0.75 18.2874
Quantile 0.975 18.5959
Mean 19.8984
Stdev 0.195347
Quantile 0.025 19.4718
Quantile 0.25 19.8026
Quantile 0.5 19.898
Quantile 0.75 19.9932
Quantile 0.975 20.3236
Mean 23.0947
Stdev 0.227004
Quantile 0.025 22.5938
Quantile 0.25 22.9844
Quantile 0.5 23.0957
Quantile 0.75 23.2057
Quantile 0.975 23.5837
Mean 17.5008
Stdev 0.17194
Quantile 0.025 17.1286
Quantile 0.25 17.4159
Quantile 0.5 17.4995
Quantile 0.75 17.5835
Quantile 0.975 17.8784
Mean 20.1954
Stdev 0.198759
Quantile 0.025 19.7568
Quantile 0.25 20.0989
Quantile 0.5 20.1962
Quantile 0.75 20.2925
Quantile 0.975 20.6237
Mean 18.2026
Stdev 0.178784
Quantile 0.025 17.8184
Quantile 0.25 18.1138
Quantile 0.5 18.2006
Quantile 0.75 18.2881
Quantile 0.975 18.598
Mean 13.605
Stdev 0.133939
Quantile 0.025 13.322
Quantile 0.25 13.5378
Quantile 0.5 13.6022
Quantile 0.75 13.6682
Quantile 0.975 13.9059
Mean 19.5924
Stdev 0.192892
Quantile 0.025 19.1622
Quantile 0.25 19.4996
Quantile 0.5 19.5945
Quantile 0.75 19.6876
Quantile 0.975 20.0033
Mean 15.205
Stdev 0.14965
Quantile 0.025 14.8877
Quantile 0.25 15.13
Quantile 0.5 15.2021
Quantile 0.75 15.2757
Quantile 0.975 15.5402
Mean 14.5003
Stdev 0.142631
Quantile 0.025 14.1912
Quantile 0.25 14.4301
Quantile 0.5 14.4994
Quantile 0.75 14.569
Quantile 0.975 14.8132
Mean 15.6009
Stdev 0.153469
Quantile 0.025 15.2691
Quantile 0.25 15.5252
Quantile 0.5 15.5997
Quantile 0.75 15.6747
Quantile 0.975 15.9384
Mean 13.9045
Stdev 0.136847
Quantile 0.025 13.6143
Quantile 0.25 13.8359
Quantile 0.5 13.9019
Quantile 0.75 13.9692
Quantile 0.975 14.2109
Mean 16.6008
Stdev 0.163408
Quantile 0.025 16.2473
Quantile 0.25 16.5202
Quantile 0.5 16.5996
Quantile 0.75 16.6793
Quantile 0.975 16.96
Mean 14.8078
Stdev 0.145952
Quantile 0.025 14.503
Quantile 0.25 14.7339
Quantile 0.5 14.8038
Quantile 0.75 14.8759
Quantile 0.975 15.1391
Mean 18.4083
Stdev 0.181557
Quantile 0.025 18.0268
Quantile 0.25 18.3168
Quantile 0.5 18.4039
Quantile 0.75 18.4933
Quantile 0.975 18.8183
Mean 20.9937
Stdev 0.206508
Quantile 0.025 20.5357
Quantile 0.25 20.8938
Quantile 0.5 20.9952
Quantile 0.75 21.095
Quantile 0.975 21.4362
Mean 12.7065
Stdev 0.12515
Quantile 0.025 12.4449
Quantile 0.25 12.6432
Quantile 0.5 12.7032
Quantile 0.75 12.765
Quantile 0.975 12.9903
Mean 14.5052
Stdev 0.142975
Quantile 0.025 14.2029
Quantile 0.25 14.4335
Quantile 0.5 14.5023
Quantile 0.75 14.5726
Quantile 0.975 14.8263
Mean 13.2072
Stdev 0.130119
Quantile 0.025 12.9359
Quantile 0.25 13.1413
Quantile 0.5 13.2035
Quantile 0.75 13.2679
Quantile 0.975 13.5029
Mean 13.101
Stdev 0.128968
Quantile 0.025 12.8226
Quantile 0.25 13.0373
Quantile 0.5 13.0999
Quantile 0.75 13.1629
Quantile 0.975 13.3851
Mean 13.5094
Stdev 0.133448
Quantile 0.025 13.2342
Quantile 0.25 13.4413
Quantile 0.5 13.5048
Quantile 0.75 13.5709
Quantile 0.975 13.8155
Mean 18.9043
Stdev 0.185634
Quantile 0.025 18.5078
Quantile 0.25 18.8116
Quantile 0.5 18.9015
Quantile 0.75 18.9927
Quantile 0.975 19.3171
Mean 20.0046
Stdev 0.196783
Quantile 0.025 19.5843
Quantile 0.25 19.9065
Quantile 0.5 20.0016
Quantile 0.75 20.0981
Quantile 0.975 20.4423
Mean 21.003
Stdev 0.206408
Quantile 0.025 20.5594
Quantile 0.25 20.9005
Quantile 0.5 21.0006
Quantile 0.75 21.1017
Quantile 0.975 21.4594
Mean 24.6881
Stdev 0.242697
Quantile 0.025 24.1433
Quantile 0.25 24.5718
Quantile 0.5 24.6917
Quantile 0.75 24.8086
Quantile 0.975 25.2012
Mean 30.7763
Stdev 0.302606
Quantile 0.025 30.0843
Quantile 0.25 30.6336
Quantile 0.5 30.7844
Quantile 0.75 30.9291
Quantile 0.975 31.4023
Mean 34.8722
Stdev 0.343339
Quantile 0.025 34.0855
Quantile 0.25 34.7107
Quantile 0.5 34.8818
Quantile 0.75 35.0457
Quantile 0.975 35.581
Mean 26.6069
Stdev 0.261935
Quantile 0.025 26.0487
Quantile 0.25 26.4762
Quantile 0.5 26.6026
Quantile 0.75 26.7312
Quantile 0.975 27.1909
Mean 25.3011
Stdev 0.249045
Quantile 0.025 24.762
Quantile 0.25 25.1783
Quantile 0.5 25.2993
Quantile 0.75 25.4208
Quantile 0.975 25.8482
Mean 24.6957
Stdev 0.242582
Quantile 0.025 24.1625
Quantile 0.25 24.5774
Quantile 0.5 24.6961
Quantile 0.75 24.814
Quantile 0.975 25.2203
Mean 21.1996
Stdev 0.208019
Quantile 0.025 20.7474
Quantile 0.25 21.0972
Quantile 0.5 21.1986
Quantile 0.75 21.3002
Quantile 0.975 21.6544
Mean 19.3047
Stdev 0.190033
Quantile 0.025 18.8993
Quantile 0.25 19.21
Quantile 0.5 19.3017
Quantile 0.75 19.395
Quantile 0.975 19.728
Mean 19.9976
Stdev 0.19655
Quantile 0.025 19.5672
Quantile 0.25 19.9015
Quantile 0.5 19.9975
Quantile 0.75 20.0931
Quantile 0.975 20.4243
Mean 16.6095
Stdev 0.163984
Quantile 0.025 16.2682
Quantile 0.25 16.5264
Quantile 0.5 16.6047
Quantile 0.75 16.6857
Quantile 0.975 16.9829
Mean 14.4082
Stdev 0.142154
Quantile 0.025 14.1123
Quantile 0.25 14.3361
Quantile 0.5 14.404
Quantile 0.75 14.4743
Quantile 0.975 14.7318
Mean 19.395
Stdev 0.190889
Quantile 0.025 18.973
Quantile 0.25 19.3025
Quantile 0.5 19.396
Quantile 0.75 19.4885
Quantile 0.975 19.8055
Mean 19.7062
Stdev 0.193708
Quantile 0.025 19.2953
Quantile 0.25 19.6092
Quantile 0.5 19.7026
Quantile 0.75 19.7979
Quantile 0.975 20.1397
Mean 20.5072
Stdev 0.201987
Quantile 0.025 20.0796
Quantile 0.25 20.4059
Quantile 0.5 20.5031
Quantile 0.75 20.6024
Quantile 0.975 20.9603
Mean 25.0028
Stdev 0.245691
Quantile 0.025 24.4736
Quantile 0.25 24.881
Quantile 0.5 25.0003
Quantile 0.75 25.1205
Quantile 0.975 25.5449
Mean 23.3967
Stdev 0.229823
Quantile 0.025 22.8928
Quantile 0.25 23.2844
Quantile 0.5 23.3968
Quantile 0.75 23.5086
Quantile 0.975 23.895
Mean 18.9127
Stdev 0.186785
Quantile 0.025 18.5269
Quantile 0.25 18.8175
Quantile 0.5 18.9065
Quantile 0.75 18.999
Quantile 0.975 19.3407
Mean 35.3872
Stdev 0.34815
Quantile 0.025 34.6119
Quantile 0.25 35.2194
Quantile 0.5 35.3906
Quantile 0.75 35.5587
Quantile 0.975 36.1301
Mean 24.7069
Stdev 0.243577
Quantile 0.025 24.1885
Quantile 0.25 24.5853
Quantile 0.5 24.7027
Quantile 0.75 24.8222
Quantile 0.975 25.2507
Mean 31.5866
Stdev 0.310439
Quantile 0.025 30.8925
Quantile 0.25 31.4374
Quantile 0.5 31.5905
Quantile 0.75 31.7402
Quantile 0.975 32.2459
Mean 23.2945
Stdev 0.228832
Quantile 0.025 22.7893
Quantile 0.25 23.1833
Quantile 0.5 23.2955
Quantile 0.75 23.4065
Quantile 0.975 23.7871
Mean 19.6004
Stdev 0.192675
Quantile 0.025 19.1826
Quantile 0.25 19.5054
Quantile 0.5 19.5992
Quantile 0.75 19.6932
Quantile 0.975 20.0229
Mean 18.6996
Stdev 0.184043
Quantile 0.025 18.2994
Quantile 0.25 18.6092
Quantile 0.5 18.6988
Quantile 0.75 18.7884
Quantile 0.975 19.1021
Mean 16.009
Stdev 0.157834
Quantile 0.025 15.6802
Quantile 0.25 15.9289
Quantile 0.5 16.0044
Quantile 0.75 16.0824
Quantile 0.975 16.368
Mean 22.2005
Stdev 0.218373
Quantile 0.025 21.7271
Quantile 0.25 22.0929
Quantile 0.5 22.1991
Quantile 0.75 22.3056
Quantile 0.975 22.6795
Mean 25.0083
Stdev 0.246458
Quantile 0.025 24.486
Quantile 0.25 24.8849
Quantile 0.5 25.0036
Quantile 0.75 25.1247
Quantile 0.975 25.5606
Mean 32.9961
Stdev 0.324942
Quantile 0.025 32.2844
Quantile 0.25 32.8374
Quantile 0.5 32.9959
Quantile 0.75 33.1537
Quantile 0.975 33.7018
Mean 23.5025
Stdev 0.231229
Quantile 0.025 23.0043
Quantile 0.25 23.3881
Quantile 0.5 23.5002
Quantile 0.75 23.6133
Quantile 0.975 24.0127
Mean 19.4052
Stdev 0.191172
Quantile 0.025 18.9982
Quantile 0.25 19.3098
Quantile 0.5 19.402
Quantile 0.75 19.4958
Quantile 0.975 19.8318
Mean 21.9948
Stdev 0.216065
Quantile 0.025 21.5179
Quantile 0.25 21.8898
Quantile 0.5 21.9958
Quantile 0.75 22.1006
Quantile 0.975 22.46
Mean 17.404
Stdev 0.171659
Quantile 0.025 17.0373
Quantile 0.25 17.3186
Quantile 0.5 17.4014
Quantile 0.75 17.4855
Quantile 0.975 17.786
Mean 20.9003
Stdev 0.205581
Quantile 0.025 20.4544
Quantile 0.25 20.7991
Quantile 0.5 20.8991
Quantile 0.75 20.9993
Quantile 0.975 21.351
Mean 24.2018
Stdev 0.238403
Quantile 0.025 23.6868
Quantile 0.25 24.0841
Quantile 0.5 24.1998
Quantile 0.75 24.3161
Quantile 0.975 24.7267
Mean 21.6987
Stdev 0.213026
Quantile 0.025 21.2343
Quantile 0.25 21.5941
Quantile 0.5 21.6981
Quantile 0.75 21.8019
Quantile 0.975 22.1632
Mean 22.802
Stdev 0.224469
Quantile 0.025 22.3175
Quantile 0.25 22.691
Quantile 0.5 22.7999
Quantile 0.75 22.9096
Quantile 0.975 23.2965
Mean 23.4104
Stdev 0.230587
Quantile 0.025 22.9258
Quantile 0.25 23.2942
Quantile 0.5 23.4048
Quantile 0.75 23.5185
Quantile 0.975 23.9309
Mean 24.1052
Stdev 0.237109
Quantile 0.025 23.5983
Quantile 0.25 23.9871
Quantile 0.5 24.1018
Quantile 0.75 24.218
Quantile 0.975 24.6322
Mean 21.4053
Stdev 0.210716
Quantile 0.025 20.956
Quantile 0.25 21.3002
Quantile 0.5 21.402
Quantile 0.75 21.5054
Quantile 0.975 21.8748
Mean 20.0078
Stdev 0.197121
Quantile 0.025 19.5918
Quantile 0.25 19.9088
Quantile 0.5 20.0035
Quantile 0.75 20.1005
Quantile 0.975 20.4512
Mean 20.8046
Stdev 0.204647
Quantile 0.025 20.3673
Quantile 0.25 20.7026
Quantile 0.5 20.8016
Quantile 0.75 20.902
Quantile 0.975 21.2596
Mean 21.2063
Stdev 0.20855
Quantile 0.025 20.7631
Quantile 0.25 21.1019
Quantile 0.5 21.2025
Quantile 0.75 21.3051
Quantile 0.975 21.6723
Mean 20.3035
Stdev 0.199801
Quantile 0.025 19.875
Quantile 0.25 20.2042
Quantile 0.5 20.301
Quantile 0.75 20.3988
Quantile 0.975 20.7463
Mean 27.9935
Stdev 0.274993
Quantile 0.025 27.3865
Quantile 0.25 27.8598
Quantile 0.5 27.9947
Quantile 0.75 28.128
Quantile 0.975 28.5856
Mean 23.9077
Stdev 0.235599
Quantile 0.025 23.4081
Quantile 0.25 23.7898
Quantile 0.5 23.9033
Quantile 0.75 24.019
Quantile 0.975 24.4354
Mean 24.7962
Stdev 0.243867
Quantile 0.025 24.2608
Quantile 0.25 24.6772
Quantile 0.5 24.7964
Quantile 0.75 24.9148
Quantile 0.975 25.3244
Mean 22.904
Stdev 0.224862
Quantile 0.025 22.422
Quantile 0.25 22.7921
Quantile 0.5 22.9011
Quantile 0.75 23.0114
Quantile 0.975 23.4023
Mean 23.9084
Stdev 0.235488
Quantile 0.025 23.4099
Quantile 0.25 23.7903
Quantile 0.5 23.9036
Quantile 0.75 24.0194
Quantile 0.975 24.4366
Mean 26.5988
Stdev 0.261287
Quantile 0.025 26.0296
Quantile 0.25 26.4704
Quantile 0.5 26.5978
Quantile 0.75 26.7252
Quantile 0.975 27.169
Mean 22.5056
Stdev 0.221409
Quantile 0.025 22.0334
Quantile 0.25 22.3951
Quantile 0.5 22.5021
Quantile 0.75 22.6108
Quantile 0.975 22.9988
Mean 22.2014
Stdev 0.218266
Quantile 0.025 21.7297
Quantile 0.25 22.0936
Quantile 0.5 22.1996
Quantile 0.75 22.3063
Quantile 0.975 22.6815
Mean 23.6097
Stdev 0.232796
Quantile 0.025 23.1193
Quantile 0.25 23.4927
Quantile 0.5 23.6045
Quantile 0.75 23.7191
Quantile 0.975 24.1342
Mean 28.6979
Stdev 0.282247
Quantile 0.025 28.0817
Quantile 0.25 28.5596
Quantile 0.5 28.6972
Quantile 0.75 28.8346
Quantile 0.975 29.3127
Mean 22.607
Stdev 0.222213
Quantile 0.025 22.1352
Quantile 0.25 22.4957
Quantile 0.5 22.6029
Quantile 0.75 22.7122
Quantile 0.975 23.1039
Mean 22.0005
Stdev 0.216016
Quantile 0.025 21.5324
Quantile 0.25 21.894
Quantile 0.5 21.9991
Quantile 0.75 22.1047
Quantile 0.975 22.4743
Mean 22.9029
Stdev 0.224934
Quantile 0.025 22.4189
Quantile 0.25 22.7913
Quantile 0.5 22.9005
Quantile 0.75 23.0106
Quantile 0.975 23.3996
Mean 24.9889
Stdev 0.245615
Quantile 0.025 24.4389
Quantile 0.25 24.8709
Quantile 0.5 24.9921
Quantile 0.75 25.1106
Quantile 0.975 25.5096
Mean 20.6115
Stdev 0.203212
Quantile 0.025 20.1882
Quantile 0.25 20.5085
Quantile 0.5 20.6057
Quantile 0.75 20.7062
Quantile 0.975 21.0737
Mean 28.3903
Stdev 0.279472
Quantile 0.025 27.7688
Quantile 0.25 28.2555
Quantile 0.5 28.3928
Quantile 0.75 28.5278
Quantile 0.975 28.9876
Mean 21.4045
Stdev 0.210673
Quantile 0.025 20.954
Quantile 0.25 21.2997
Quantile 0.5 21.4015
Quantile 0.75 21.5048
Quantile 0.975 21.8727
Mean 38.6968
Stdev 0.380366
Quantile 0.025 37.866
Quantile 0.25 38.5104
Quantile 0.5 38.696
Quantile 0.75 38.8812
Quantile 0.975 39.5248
Mean 43.7756
Stdev 0.430515
Quantile 0.025 42.8044
Quantile 0.25 43.5703
Quantile 0.5 43.7833
Quantile 0.75 43.9903
Quantile 0.975 44.6808
Mean 33.1978
Stdev 0.326106
Quantile 0.025 32.4864
Quantile 0.25 33.0378
Quantile 0.5 33.1969
Quantile 0.75 33.3558
Quantile 0.975 33.9085
Mean 27.4756
Stdev 0.27017
Quantile 0.025 26.8533
Quantile 0.25 27.349
Quantile 0.5 27.4841
Quantile 0.75 27.613
Quantile 0.975 28.0294
Mean 26.5018
Stdev 0.260237
Quantile 0.025 25.9396
Quantile 0.25 26.3731
Quantile 0.5 26.4997
Quantile 0.75 26.6269
Quantile 0.975 27.0743
Mean 18.6158
Stdev 0.184229
Quantile 0.025 18.2406
Quantile 0.25 18.5211
Quantile 0.5 18.6083
Quantile 0.75 18.6999
Quantile 0.975 19.0427
Mean 19.3061
Stdev 0.189997
Quantile 0.025 18.9029
Quantile 0.25 19.2109
Quantile 0.5 19.3025
Quantile 0.75 19.3959
Quantile 0.975 19.7312
Mean 20.1045
Stdev 0.197643
Quantile 0.025 19.6823
Quantile 0.25 20.006
Quantile 0.5 20.1016
Quantile 0.75 20.1986
Quantile 0.975 20.544
Mean 19.4909
Stdev 0.191709
Quantile 0.025 19.0609
Quantile 0.25 19.399
Quantile 0.5 19.4936
Quantile 0.75 19.5859
Quantile 0.975 19.8967
Mean 19.4987
Stdev 0.191772
Quantile 0.025 19.0802
Quantile 0.25 19.4047
Quantile 0.5 19.4982
Quantile 0.75 19.5915
Quantile 0.975 19.9166
Mean 20.3957
Stdev 0.200349
Quantile 0.025 19.9541
Quantile 0.25 20.2982
Quantile 0.5 20.3964
Quantile 0.75 20.4936
Quantile 0.975 20.8278
Mean 19.8091
Stdev 0.19514
Quantile 0.025 19.3995
Quantile 0.25 19.7107
Quantile 0.5 19.8043
Quantile 0.75 19.9006
Quantile 0.975 20.2501
Mean 19.4045
Stdev 0.19126
Quantile 0.025 18.9962
Quantile 0.25 19.3093
Quantile 0.5 19.4016
Quantile 0.75 19.4953
Quantile 0.975 19.8303
Mean 21.6963
Stdev 0.213381
Quantile 0.025 21.2273
Quantile 0.25 21.5923
Quantile 0.5 21.6967
Quantile 0.75 21.8002
Quantile 0.975 22.1579
Mean 22.8061
Stdev 0.224383
Quantile 0.025 22.3282
Quantile 0.25 22.694
Quantile 0.5 22.8023
Quantile 0.75 22.9125
Quantile 0.975 23.3065
Mean 18.799
Stdev 0.184894
Quantile 0.025 18.3961
Quantile 0.25 18.7083
Quantile 0.5 18.7984
Quantile 0.75 18.8884
Quantile 0.975 19.2024
Mean 18.704
Stdev 0.183977
Quantile 0.025 18.3106
Quantile 0.25 18.6124
Quantile 0.5 18.7013
Quantile 0.75 18.7915
Quantile 0.975 19.1128
Mean 18.5094
Stdev 0.182192
Quantile 0.025 18.1285
Quantile 0.25 18.4172
Quantile 0.5 18.5045
Quantile 0.75 18.5946
Quantile 0.975 18.9224
Mean 18.3036
Stdev 0.180027
Quantile 0.025 17.9183
Quantile 0.25 18.214
Quantile 0.5 18.3011
Quantile 0.75 18.3894
Quantile 0.975 18.7033
Mean 21.1978
Stdev 0.208478
Quantile 0.025 20.7418
Quantile 0.25 21.0958
Quantile 0.5 21.1976
Quantile 0.75 21.299
Quantile 0.975 21.6511
Mean 19.2048
Stdev 0.188939
Quantile 0.025 18.802
Quantile 0.25 19.1106
Quantile 0.5 19.2018
Quantile 0.75 19.2946
Quantile 0.975 19.6259
Mean 20.3984
Stdev 0.200618
Quantile 0.025 19.9604
Quantile 0.25 20.3001
Quantile 0.5 20.398
Quantile 0.75 20.4956
Quantile 0.975 20.8353
Mean 19.3034
Stdev 0.189844
Quantile 0.025 18.8964
Quantile 0.25 19.209
Quantile 0.5 19.301
Quantile 0.75 19.394
Quantile 0.975 19.7242
Mean 21.9949
Stdev 0.215819
Quantile 0.025 21.5186
Quantile 0.25 21.8899
Quantile 0.5 21.9958
Quantile 0.75 22.1006
Quantile 0.975 22.4595
Mean 20.2983
Stdev 0.199388
Quantile 0.025 19.8628
Quantile 0.25 20.2006
Quantile 0.5 20.2979
Quantile 0.75 20.395
Quantile 0.975 20.7323
Mean 20.4966
Stdev 0.201849
Quantile 0.025 20.0532
Quantile 0.25 20.3983
Quantile 0.5 20.4969
Quantile 0.75 20.5948
Quantile 0.975 20.9336
Mean 17.3048
Stdev 0.170267
Quantile 0.025 16.9425
Quantile 0.25 17.2197
Quantile 0.5 17.3019
Quantile 0.75 17.3855
Quantile 0.975 17.6848
Mean 18.802
Stdev 0.184984
Quantile 0.025 18.4034
Quantile 0.25 18.7105
Quantile 0.5 18.8002
Quantile 0.75 18.8906
Quantile 0.975 19.2102
Mean 21.3869
Stdev 0.210649
Quantile 0.025 20.9099
Quantile 0.25 21.2868
Quantile 0.5 21.3912
Quantile 0.75 21.4922
Quantile 0.975 21.828
Mean 15.6977
Stdev 0.154106
Quantile 0.025 15.3596
Quantile 0.25 15.6224
Quantile 0.5 15.6978
Quantile 0.75 15.7727
Quantile 0.975 16.0315
Mean 16.1958
Stdev 0.159024
Quantile 0.025 15.8442
Quantile 0.25 16.1186
Quantile 0.5 16.1967
Quantile 0.75 16.2738
Quantile 0.975 16.5376
Mean 18.0023
Stdev 0.177018
Quantile 0.025 17.6215
Quantile 0.25 17.9146
Quantile 0.5 18.0004
Quantile 0.75 18.087
Quantile 0.975 18.3934
Mean 14.3048
Stdev 0.140885
Quantile 0.025 14.0063
Quantile 0.25 14.2342
Quantile 0.5 14.302
Quantile 0.75 14.3713
Quantile 0.975 14.6205
Mean 19.207
Stdev 0.188966
Quantile 0.025 18.8075
Quantile 0.25 19.1121
Quantile 0.5 19.2031
Quantile 0.75 19.2961
Quantile 0.975 19.6312
Mean 19.6002
Stdev 0.192915
Quantile 0.025 19.1816
Quantile 0.25 19.5053
Quantile 0.5 19.5991
Quantile 0.75 19.6931
Quantile 0.975 20.023
Mean 22.9894
Stdev 0.226114
Quantile 0.025 22.4824
Quantile 0.25 22.8809
Quantile 0.5 22.9925
Quantile 0.75 23.1014
Quantile 0.975 23.4682
Mean 18.4022
Stdev 0.180836
Quantile 0.025 18.013
Quantile 0.25 18.3126
Quantile 0.5 18.4003
Quantile 0.75 18.4888
Quantile 0.975 18.8015
Mean 15.6071
Stdev 0.153835
Quantile 0.025 15.284
Quantile 0.25 15.5295
Quantile 0.5 15.6033
Quantile 0.75 15.6792
Quantile 0.975 15.9546
Mean 18.1088
Stdev 0.178427
Quantile 0.025 17.735
Quantile 0.25 18.0187
Quantile 0.5 18.1042
Quantile 0.75 18.1923
Quantile 0.975 18.5127
Mean 17.4096
Stdev 0.171427
Quantile 0.025 17.0522
Quantile 0.25 17.3226
Quantile 0.5 17.4047
Quantile 0.75 17.4895
Quantile 0.975 17.7992
Mean 17.1047
Stdev 0.168407
Quantile 0.025 16.7464
Quantile 0.25 17.0206
Quantile 0.5 17.1019
Quantile 0.75 17.1845
Quantile 0.975 17.4807
Mean 13.3259
Stdev 0.134518
Quantile 0.025 13.0749
Quantile 0.25 13.2533
Quantile 0.5 13.3144
Quantile 0.75 13.3828
Quantile 0.975 13.657
Mean 17.7954
Stdev 0.175032
Quantile 0.025 17.4085
Quantile 0.25 17.7105
Quantile 0.5 17.7964
Quantile 0.75 17.8811
Quantile 0.975 18.1718
Mean 14.0097
Stdev 0.138307
Quantile 0.025 13.7244
Quantile 0.25 13.9391
Quantile 0.5 14.0049
Quantile 0.75 14.0735
Quantile 0.975 14.3269
Mean 14.3936
Stdev 0.141486
Quantile 0.025 14.0768
Quantile 0.25 14.3256
Quantile 0.5 14.3954
Quantile 0.75 14.4637
Quantile 0.975 14.6936
Mean 13.419
Stdev 0.133739
Quantile 0.025 13.1587
Quantile 0.25 13.3484
Quantile 0.5 13.4105
Quantile 0.75 13.4778
Quantile 0.975 13.7393
Mean 15.6006
Stdev 0.153459
Quantile 0.025 15.2684
Quantile 0.25 15.525
Quantile 0.5 15.5995
Quantile 0.75 15.6744
Quantile 0.975 15.9376
Mean 11.8025
Stdev 0.116243
Quantile 0.025 11.554
Quantile 0.25 11.7447
Quantile 0.5 11.8008
Quantile 0.75 11.8578
Quantile 0.975 12.0609
Mean 13.8152
Stdev 0.137131
Quantile 0.025 13.5414
Quantile 0.25 13.7438
Quantile 0.5 13.8082
Quantile 0.75 13.8768
Quantile 0.975 14.1378
Mean 15.6038
Stdev 0.1536
Quantile 0.025 15.2761
Quantile 0.25 15.5272
Quantile 0.5 15.6014
Quantile 0.75 15.6767
Quantile 0.975 15.9459
Mean 14.5943
Stdev 0.143507
Quantile 0.025 14.274
Quantile 0.25 14.5252
Quantile 0.5 14.5958
Quantile 0.75 14.6651
Quantile 0.975 14.8997
Mean 17.7869
Stdev 0.175234
Quantile 0.025 17.3869
Quantile 0.25 17.7042
Quantile 0.5 17.7914
Quantile 0.75 17.8751
Quantile 0.975 18.1504
Mean 15.4061
Stdev 0.151692
Quantile 0.025 15.0861
Quantile 0.25 15.3298
Quantile 0.5 15.4027
Quantile 0.75 15.4775
Quantile 0.975 15.7473
Mean 21.4966
Stdev 0.211284
Quantile 0.025 21.0328
Quantile 0.25 21.3935
Quantile 0.5 21.4969
Quantile 0.75 21.5995
Quantile 0.975 21.9542
Mean 19.5774
Stdev 0.193401
Quantile 0.025 19.1246
Quantile 0.25 19.4884
Quantile 0.5 19.5857
Quantile 0.75 19.6771
Quantile 0.975 19.9659
Mean 15.3115
Stdev 0.150921
Quantile 0.025 15.0017
Quantile 0.25 15.2341
Quantile 0.5 15.3059
Quantile 0.75 15.381
Quantile 0.975 15.6588
Mean 19.3901
Stdev 0.190875
Quantile 0.025 18.9608
Quantile 0.25 19.2989
Quantile 0.5 19.3931
Quantile 0.75 19.4849
Quantile 0.975 19.793
Mean 17.0065
Stdev 0.167446
Quantile 0.025 16.6531
Quantile 0.25 16.9224
Quantile 0.5 17.0029
Quantile 0.75 17.0854
Quantile 0.975 17.3831
Mean 15.6128
Stdev 0.154447
Quantile 0.025 15.2974
Quantile 0.25 15.5335
Quantile 0.5 15.6067
Quantile 0.75 15.6834
Quantile 0.975 15.9699
Mean 13.1091
Stdev 0.129582
Quantile 0.025 12.8419
Quantile 0.25 13.043
Quantile 0.5 13.1047
Quantile 0.75 13.1689
Quantile 0.975 13.4065
Mean 41.256
Stdev 0.408044
Quantile 0.025 40.3053
Quantile 0.25 41.0675
Quantile 0.5 41.2719
Quantile 0.75 41.465
Quantile 0.975 42.0813
Mean 24.3024
Stdev 0.238804
Quantile 0.025 23.7876
Quantile 0.25 24.1842
Quantile 0.5 24.3001
Quantile 0.75 24.4169
Quantile 0.975 24.8289
Mean 23.3044
Stdev 0.229347
Quantile 0.025 22.813
Quantile 0.25 23.1904
Quantile 0.5 23.3013
Quantile 0.75 23.4137
Quantile 0.975 23.8132
Mean 26.9744
Stdev 0.266112
Quantile 0.025 26.359
Quantile 0.25 26.8505
Quantile 0.5 26.9835
Quantile 0.75 27.11
Quantile 0.975 27.5176
Mean 49.9801
Stdev 0.492095
Quantile 0.025 48.8813
Quantile 0.25 49.7435
Quantile 0.5 49.9856
Quantile 0.75 50.2229
Quantile 0.975 51.0271
Mean 49.9493
Stdev 0.493154
Quantile 0.025 48.8038
Quantile 0.25 49.7205
Quantile 0.5 49.9675
Quantile 0.75 50.2013
Quantile 0.975 50.9504
Mean 49.982
Stdev 0.491191
Quantile 0.025 48.8883
Quantile 0.25 49.745
Quantile 0.5 49.9867
Quantile 0.75 50.2241
Quantile 0.975 51.0299
Mean 22.7084
Stdev 0.223561
Quantile 0.025 22.2359
Quantile 0.25 22.5961
Quantile 0.5 22.7037
Quantile 0.75 22.8138
Quantile 0.975 23.2105
Mean 25.0051
Stdev 0.246264
Quantile 0.025 24.4781
Quantile 0.25 24.8826
Quantile 0.5 25.0017
Quantile 0.75 25.1223
Quantile 0.975 25.5521
Mean 49.9634
Stdev 0.491678
Quantile 0.025 48.8413
Quantile 0.25 49.7312
Quantile 0.5 49.9758
Quantile 0.75 50.211
Quantile 0.975 50.9833
Mean 23.7982
Stdev 0.233515
Quantile 0.025 23.2885
Quantile 0.25 23.6836
Quantile 0.5 23.7977
Quantile 0.75 23.9115
Quantile 0.975 24.3067
Mean 23.7971
Stdev 0.233894
Quantile 0.025 23.2849
Quantile 0.25 23.6827
Quantile 0.5 23.797
Quantile 0.75 23.9108
Quantile 0.975 24.3049
Mean 22.3036
Stdev 0.219618
Quantile 0.025 21.8322
Quantile 0.25 22.1946
Quantile 0.5 22.3009
Quantile 0.75 22.4084
Quantile 0.975 22.7901
Mean 17.419
Stdev 0.172873
Quantile 0.025 17.0735
Quantile 0.25 17.3291
Quantile 0.5 17.4102
Quantile 0.75 17.4967
Quantile 0.975 17.8255
Mean 19.1008
Stdev 0.18755
Quantile 0.025 18.6948
Quantile 0.25 19.0082
Quantile 0.5 19.0994
Quantile 0.75 19.1911
Quantile 0.975 19.5126
Mean 23.09
Stdev 0.227222
Quantile 0.025 22.5815
Quantile 0.25 22.9809
Quantile 0.5 23.0929
Quantile 0.75 23.2024
Quantile 0.975 23.5722
Mean 23.605
Stdev 0.232332
Quantile 0.025 23.1083
Quantile 0.25 23.4894
Quantile 0.5 23.6017
Quantile 0.75 23.7156
Quantile 0.975 24.1214
Mean 22.5928
Stdev 0.222109
Quantile 0.025 22.0996
Quantile 0.25 22.4854
Quantile 0.5 22.5946
Quantile 0.75 22.702
Quantile 0.975 23.0681
Mean 29.386
Stdev 0.288872
Quantile 0.025 28.7378
Quantile 0.25 29.2475
Quantile 0.5 29.3902
Quantile 0.75 29.5294
Quantile 0.975 29.9971
Mean 23.2133
Stdev 0.228899
Quantile 0.025 22.737
Quantile 0.25 23.0972
Quantile 0.5 23.2066
Quantile 0.75 23.3198
Quantile 0.975 23.7344
Mean 24.6123
Stdev 0.242688
Quantile 0.025 24.1045
Quantile 0.25 24.4897
Quantile 0.5 24.6059
Quantile 0.75 24.7257
Quantile 0.975 25.1622
Mean 29.9032
Stdev 0.294022
Quantile 0.025 29.2698
Quantile 0.25 29.7576
Quantile 0.5 29.9003
Quantile 0.75 30.0441
Quantile 0.975 30.5519
Mean 37.1746
Stdev 0.365881
Quantile 0.025 36.3422
Quantile 0.25 37.0013
Quantile 0.5 37.183
Quantile 0.75 37.3583
Quantile 0.975 37.9365
Mean 39.8133
Stdev 0.391428
Quantile 0.025 38.984
Quantile 0.25 39.617
Quantile 0.5 39.8057
Quantile 0.75 39.9984
Quantile 0.975 40.6903
Mean 36.1622
Stdev 0.357575
Quantile 0.025 35.3301
Quantile 0.25 35.9968
Quantile 0.5 36.1759
Quantile 0.75 36.3451
Quantile 0.975 36.8866
Mean 37.9006
Stdev 0.373043
Quantile 0.025 37.0915
Quantile 0.25 37.717
Quantile 0.5 37.8983
Quantile 0.75 38.0802
Quantile 0.975 38.7185
Mean 32.4991
Stdev 0.319448
Quantile 0.025 31.8042
Quantile 0.25 32.3421
Quantile 0.5 32.4978
Quantile 0.75 32.6535
Quantile 0.975 33.1973
Mean 26.3934
Stdev 0.259767
Quantile 0.025 25.8194
Quantile 0.25 26.2675
Quantile 0.5 26.3947
Quantile 0.75 26.5205
Quantile 0.975 26.9523
Mean 29.5944
Stdev 0.290705
Quantile 0.025 28.9546
Quantile 0.25 29.4527
Quantile 0.5 29.5951
Quantile 0.75 29.7363
Quantile 0.975 30.2223
Mean 49.9642
Stdev 0.491903
Quantile 0.025 48.8428
Quantile 0.25 49.7318
Quantile 0.5 49.9762
Quantile 0.75 50.2116
Quantile 0.975 50.986
Mean 32.0052
Stdev 0.314744
Quantile 0.025 31.3297
Quantile 0.25 31.8489
Quantile 0.5 32.0013
Quantile 0.75 32.1555
Quantile 0.975 32.7022
Mean 29.8013
Stdev 0.29334
Quantile 0.025 29.1663
Quantile 0.25 29.6567
Quantile 0.5 29.7992
Quantile 0.75 29.9423
Quantile 0.975 30.4457
Mean 34.9065
Stdev 0.343103
Quantile 0.025 34.1714
Quantile 0.25 34.7358
Quantile 0.5 34.9019
Quantile 0.75 35.0701
Quantile 0.975 35.6674
Mean 36.9771
Stdev 0.363774
Quantile 0.025 36.1531
Quantile 0.25 36.8042
Quantile 0.5 36.9846
Quantile 0.75 37.1591
Quantile 0.975 37.7384
Mean 30.5066
Stdev 0.300666
Quantile 0.025 29.8639
Quantile 0.25 30.3571
Quantile 0.5 30.5023
Quantile 0.75 30.6495
Quantile 0.975 31.1752
Mean 36.3951
Stdev 0.35794
Quantile 0.025 35.6103
Quantile 0.25 36.2203
Quantile 0.5 36.3952
Quantile 0.75 36.5691
Quantile 0.975 37.1714
Mean 31.0975
Stdev 0.306043
Quantile 0.025 30.429
Quantile 0.25 30.9476
Quantile 0.5 31.0968
Quantile 0.75 31.2457
Quantile 0.975 31.7638
Mean 29.1018
Stdev 0.287075
Quantile 0.025 28.481
Quantile 0.25 28.9603
Quantile 0.5 29.0995
Quantile 0.75 29.2394
Quantile 0.975 29.7334
Mean 49.9534
Stdev 0.492198
Quantile 0.025 48.8162
Quantile 0.25 49.7237
Quantile 0.5 49.9699
Quantile 0.75 50.2041
Quantile 0.975 50.959
Mean 33.3122
Stdev 0.328143
Quantile 0.025 32.6184
Quantile 0.25 33.1475
Quantile 0.5 33.3054
Quantile 0.75 33.4668
Quantile 0.975 34.049
Mean 30.3081
Stdev 0.298779
Quantile 0.025 29.6717
Quantile 0.25 30.1591
Quantile 0.5 30.3031
Quantile 0.75 30.4496
Quantile 0.975 30.9747
Mean 34.6034
Stdev 0.340887
Quantile 0.025 33.8682
Quantile 0.25 34.4349
Quantile 0.5 34.6001
Quantile 0.75 34.7666
Quantile 0.975 35.3551
Mean 34.8955
Stdev 0.343193
Quantile 0.025 34.1435
Quantile 0.25 34.7279
Quantile 0.5 34.8955
Quantile 0.75 35.0623
Quantile 0.975 35.6402
Mean 32.9029
Stdev 0.323907
Quantile 0.025 32.204
Quantile 0.25 32.7428
Quantile 0.5 32.8999
Quantile 0.75 33.0582
Quantile 0.975 33.6167
Mean 24.099
Stdev 0.237024
Quantile 0.025 23.5829
Quantile 0.25 23.9827
Quantile 0.5 24.0981
Quantile 0.75 24.2136
Quantile 0.975 24.6166
Mean 42.2814
Stdev 0.415109
Quantile 0.025 41.3523
Quantile 0.25 42.0819
Quantile 0.5 42.2868
Quantile 0.75 42.4872
Quantile 0.975 43.1617
Mean 48.4829
Stdev 0.476976
Quantile 0.025 47.4212
Quantile 0.25 48.2528
Quantile 0.5 48.4873
Quantile 0.75 48.7177
Quantile 0.975 49.5011
Mean 49.9513
Stdev 0.49216
Quantile 0.025 48.8114
Quantile 0.25 49.7222
Quantile 0.5 49.9687
Quantile 0.75 50.2026
Quantile 0.975 50.9536
Mean 22.6109
Stdev 0.222785
Quantile 0.025 22.1442
Quantile 0.25 22.4984
Quantile 0.5 22.6052
Quantile 0.75 22.7152
Quantile 0.975 23.1152
Mean 24.4172
Stdev 0.241077
Quantile 0.025 23.9205
Quantile 0.25 24.2941
Quantile 0.5 24.4088
Quantile 0.75 24.5284
Quantile 0.975 24.9706
Mean 22.507
Stdev 0.221502
Quantile 0.025 22.0369
Quantile 0.25 22.3961
Quantile 0.5 22.5029
Quantile 0.75 22.6118
Quantile 0.975 23.0026
Mean 24.4007
Stdev 0.239866
Quantile 0.025 23.8809
Quantile 0.25 24.2824
Quantile 0.5 24.3991
Quantile 0.75 24.5162
Quantile 0.975 24.9269
Mean 19.9995
Stdev 0.196835
Quantile 0.025 19.5713
Quantile 0.25 19.9028
Quantile 0.5 19.9986
Quantile 0.75 20.0945
Quantile 0.975 20.4298
Mean 21.7058
Stdev 0.21356
Quantile 0.025 21.251
Quantile 0.25 21.5991
Quantile 0.5 21.7022
Quantile 0.75 21.8071
Quantile 0.975 22.1822
Mean 19.3037
Stdev 0.189862
Quantile 0.025 18.8972
Quantile 0.25 19.2093
Quantile 0.5 19.3012
Quantile 0.75 19.3942
Quantile 0.975 19.7251
Mean 22.4029
Stdev 0.220027
Quantile 0.025 21.9296
Quantile 0.25 22.2937
Quantile 0.5 22.4005
Quantile 0.75 22.5082
Quantile 0.975 22.8889
Mean 28.0961
Stdev 0.276149
Quantile 0.025 27.4905
Quantile 0.25 27.9612
Quantile 0.5 28.0962
Quantile 0.75 28.2304
Quantile 0.975 28.6948
Mean 23.6946
Stdev 0.233049
Quantile 0.025 23.1803
Quantile 0.25 23.5814
Quantile 0.5 23.6955
Quantile 0.75 23.8085
Quantile 0.975 24.1967
Mean 25.0071
Stdev 0.246215
Quantile 0.025 24.4834
Quantile 0.25 24.8841
Quantile 0.5 25.0028
Quantile 0.75 25.1237
Quantile 0.975 25.5569
Mean 23.299
Stdev 0.228372
Quantile 0.025 22.8018
Quantile 0.25 23.1866
Quantile 0.5 23.2982
Quantile 0.75 23.4097
Quantile 0.975 23.7975
Mean 28.711
Stdev 0.282508
Quantile 0.025 28.1146
Quantile 0.25 28.569
Quantile 0.5 28.7049
Quantile 0.75 28.8441
Quantile 0.975 29.3461
Mean 21.5103
Stdev 0.211681
Quantile 0.025 21.0666
Quantile 0.25 21.4033
Quantile 0.5 21.5049
Quantile 0.75 21.6094
Quantile 0.975 21.9891
Mean 23.0068
Stdev 0.226683
Quantile 0.025 22.525
Quantile 0.25 22.8935
Quantile 0.5 23.0027
Quantile 0.75 23.114
Quantile 0.975 23.5135
Mean 26.7129
Stdev 0.262885
Quantile 0.025 26.1621
Quantile 0.25 26.58
Quantile 0.5 26.7061
Quantile 0.75 26.836
Quantile 0.975 27.3077
Mean 21.7043
Stdev 0.213892
Quantile 0.025 21.2464
Quantile 0.25 21.598
Quantile 0.5 21.7013
Quantile 0.75 21.8061
Quantile 0.975 22.1792
Mean 27.5087
Stdev 0.270724
Quantile 0.025 26.9342
Quantile 0.25 27.3731
Quantile 0.5 27.5036
Quantile 0.75 27.6367
Quantile 0.975 28.1145
Mean 30.092
Stdev 0.295989
Quantile 0.025 29.4371
Quantile 0.25 29.9485
Quantile 0.5 30.0937
Quantile 0.75 30.237
Quantile 0.975 30.7279
Mean 44.7912
Stdev 0.440266
Quantile 0.025 43.822
Quantile 0.25 44.5769
Quantile 0.5 44.7924
Quantile 0.75 45.0061
Quantile 0.975 45.742
Mean 50.0078
Stdev 0.491234
Quantile 0.025 48.9531
Quantile 0.25 49.7637
Quantile 0.5 50.0019
Quantile 0.75 50.2426
Quantile 0.975 51.095
Mean 37.6162
Stdev 0.370479
Quantile 0.025 36.8369
Quantile 0.25 37.4296
Quantile 0.5 37.6075
Quantile 0.75 37.7901
Quantile 0.975 38.4518
Mean 31.6087
Stdev 0.3112
Quantile 0.025 30.9463
Quantile 0.25 31.4532
Quantile 0.5 31.6034
Quantile 0.75 31.7562
Quantile 0.975 32.3032
Mean 46.6765
Stdev 0.459464
Quantile 0.025 45.6434
Quantile 0.25 46.4568
Quantile 0.5 46.6837
Quantile 0.75 46.9047
Quantile 0.975 47.6465
Mean 31.4888
Stdev 0.309407
Quantile 0.025 30.8002
Quantile 0.25 31.3395
Quantile 0.5 31.4917
Quantile 0.75 31.6413
Quantile 0.975 32.1492
Mean 24.3033
Stdev 0.239138
Quantile 0.025 23.7892
Quantile 0.25 24.1848
Quantile 0.5 24.3006
Quantile 0.75 24.4176
Quantile 0.975 24.8321
Mean 31.7142
Stdev 0.312391
Quantile 0.025 31.058
Quantile 0.25 31.5567
Quantile 0.5 31.7067
Quantile 0.75 31.8607
Quantile 0.975 32.4197
Mean 41.7181
Stdev 0.4109
Quantile 0.025 40.8539
Quantile 0.25 41.5111
Quantile 0.5 41.7084
Quantile 0.75 41.911
Quantile 0.975 42.645
Mean 48.2703
Stdev 0.47488
Quantile 0.025 47.1949
Quantile 0.25 48.0446
Quantile 0.5 48.2799
Quantile 0.75 48.5078
Quantile 0.975 49.2645
Mean 28.9953
Stdev 0.284492
Quantile 0.025 28.3707
Quantile 0.25 28.8564
Quantile 0.5 28.9957
Quantile 0.75 29.1341
Quantile 0.975 29.6111
Mean 23.9981
Stdev 0.236023
Quantile 0.025 23.4828
Quantile 0.25 23.8825
Quantile 0.5 23.9976
Quantile 0.75 24.1125
Quantile 0.975 24.5121
Mean 25.1123
Stdev 0.247438
Quantile 0.025 24.5941
Quantile 0.25 24.9873
Quantile 0.5 25.1059
Quantile 0.75 25.228
Quantile 0.975 25.6725
Mean 31.5171
Stdev 0.3105
Quantile 0.025 30.8696
Quantile 0.25 31.3598
Quantile 0.5 31.5084
Quantile 0.75 31.6619
Quantile 0.975 32.2226
Mean 23.7142
Stdev 0.233887
Quantile 0.025 23.2284
Quantile 0.25 23.5954
Quantile 0.5 23.7071
Quantile 0.75 23.8228
Quantile 0.975 24.2476
Mean 23.3116
Stdev 0.229858
Quantile 0.025 22.8306
Quantile 0.25 23.1955
Quantile 0.5 23.3056
Quantile 0.75 23.419
Quantile 0.975 23.8324
Mean 22.0337
Stdev 0.220331
Quantile 0.025 21.6086
Quantile 0.25 21.9168
Quantile 0.5 22.0186
Quantile 0.75 22.1297
Quantile 0.975 22.5647
Mean 20.1067
Stdev 0.19815
Quantile 0.025 19.6867
Quantile 0.25 20.0075
Quantile 0.5 20.1028
Quantile 0.75 20.2002
Quantile 0.975 20.5507
Mean 22.2045
Stdev 0.218144
Quantile 0.025 21.7379
Quantile 0.25 22.0958
Quantile 0.5 22.2015
Quantile 0.75 22.3085
Quantile 0.975 22.689
Mean 23.7018
Stdev 0.232747
Quantile 0.025 23.1992
Quantile 0.25 23.5867
Quantile 0.5 23.6998
Quantile 0.75 23.8136
Quantile 0.975 24.2141
Mean 17.6087
Stdev 0.173624
Quantile 0.025 17.2453
Quantile 0.25 17.521
Quantile 0.5 17.6042
Quantile 0.75 17.6899
Quantile 0.975 18.0021
Mean 18.5065
Stdev 0.182169
Quantile 0.025 18.1209
Quantile 0.25 18.4151
Quantile 0.5 18.5028
Quantile 0.75 18.5924
Quantile 0.975 18.9151
Mean 24.2898
Stdev 0.238722
Quantile 0.025 23.7562
Quantile 0.25 24.175
Quantile 0.5 24.2927
Quantile 0.75 24.4079
Quantile 0.975 24.7969
Mean 20.5078
Stdev 0.202034
Quantile 0.025 20.0811
Quantile 0.25 20.4063
Quantile 0.5 20.5035
Quantile 0.75 20.6029
Quantile 0.975 20.9619
Mean 24.5021
Stdev 0.240897
Quantile 0.025 23.9822
Quantile 0.25 24.3829
Quantile 0.5 24.4999
Quantile 0.75 24.6177
Quantile 0.975 25.0327
Mean 26.2027
Stdev 0.257476
Quantile 0.025 25.6478
Quantile 0.25 26.0752
Quantile 0.5 26.2002
Quantile 0.75 26.3262
Quantile 0.975 26.7706
Mean 24.4022
Stdev 0.240069
Quantile 0.025 23.8843
Quantile 0.25 24.2835
Quantile 0.5 24.4
Quantile 0.75 24.5173
Quantile 0.975 24.9313
Mean 24.8067
Stdev 0.244075
Quantile 0.025 24.2871
Quantile 0.25 24.6848
Quantile 0.5 24.8026
Quantile 0.75 24.9225
Quantile 0.975 25.3513
Mean 29.6005
Stdev 0.291158
Quantile 0.025 28.969
Quantile 0.25 29.4571
Quantile 0.5 29.5987
Quantile 0.75 29.7407
Quantile 0.975 30.2388
Mean 42.793
Stdev 0.420894
Quantile 0.025 41.8685
Quantile 0.25 42.5878
Quantile 0.5 42.7936
Quantile 0.75 42.998
Quantile 0.975 43.7041
Mean 21.9039
Stdev 0.215696
Quantile 0.025 21.4415
Quantile 0.25 21.7968
Quantile 0.5 21.9011
Quantile 0.75 22.0067
Quantile 0.975 22.3822
Mean 20.9053
Stdev 0.20593
Quantile 0.025 20.4662
Quantile 0.25 20.8026
Quantile 0.5 20.902
Quantile 0.75 21.003
Quantile 0.975 21.3642
Mean 43.9227
Stdev 0.438553
Quantile 0.025 42.8608
Quantile 0.25 43.7287
Quantile 0.5 43.9522
Quantile 0.75 44.1553
Quantile 0.975 44.7622
Mean 49.9826
Stdev 0.491492
Quantile 0.025 48.8892
Quantile 0.25 49.7453
Quantile 0.5 49.987
Quantile 0.75 50.2246
Quantile 0.975 51.0323
Mean 35.9923
Stdev 0.353774
Quantile 0.025 35.2124
Quantile 0.25 35.8202
Quantile 0.5 35.9935
Quantile 0.75 36.1651
Quantile 0.975 36.7551
Mean 30.1035
Stdev 0.296574
Quantile 0.025 29.4647
Quantile 0.25 29.9568
Quantile 0.5 30.1004
Quantile 0.75 30.2453
Quantile 0.975 30.7583
Mean 33.8061
Stdev 0.33228
Quantile 0.025 33.094
Quantile 0.25 33.6409
Quantile 0.5 33.8018
Quantile 0.75 33.9647
Quantile 0.975 34.5428
Mean 43.0846
Stdev 0.424142
Quantile 0.025 42.1402
Quantile 0.25 42.8802
Quantile 0.5 43.0886
Quantile 0.75 43.2934
Quantile 0.975 43.9898
Mean 48.7937
Stdev 0.479596
Quantile 0.025 47.7429
Quantile 0.25 48.5593
Quantile 0.5 48.7937
Quantile 0.75 49.0269
Quantile 0.975 49.8345
Mean 31.0216
Stdev 0.30626
Quantile 0.025 30.3903
Quantile 0.25 30.8653
Quantile 0.5 31.011
Quantile 0.75 31.1629
Quantile 0.975 31.7244
Mean 36.4813
Stdev 0.358684
Quantile 0.025 35.6744
Quantile 0.25 36.3097
Quantile 0.5 36.487
Quantile 0.75 36.6597
Quantile 0.975 37.2379
Mean 22.7901
Stdev 0.224276
Quantile 0.025 22.2881
Quantile 0.25 22.6824
Quantile 0.5 22.7929
Quantile 0.75 22.901
Quantile 0.975 23.2659
Mean 30.705
Stdev 0.30178
Quantile 0.025 30.0574
Quantile 0.25 30.5551
Quantile 0.5 30.7013
Quantile 0.75 30.8492
Quantile 0.975 31.3732
Mean 49.9784
Stdev 0.491226
Quantile 0.025 48.8793
Quantile 0.25 49.7423
Quantile 0.5 49.9846
Quantile 0.75 50.2215
Quantile 0.975 51.0208
Mean 43.4718
Stdev 0.428007
Quantile 0.025 42.5003
Quantile 0.25 43.2688
Quantile 0.5 43.4811
Quantile 0.75 43.6862
Quantile 0.975 44.3656
Mean 20.708
Stdev 0.203767
Quantile 0.025 20.278
Quantile 0.25 20.6056
Quantile 0.5 20.7036
Quantile 0.75 20.804
Quantile 0.975 21.1662
Mean 21.1002
Stdev 0.207544
Quantile 0.025 20.6498
Quantile 0.25 20.998
Quantile 0.5 21.099
Quantile 0.75 21.2002
Quantile 0.975 21.555
Mean 25.1982
Stdev 0.247671
Quantile 0.025 24.6576
Quantile 0.25 25.0767
Quantile 0.5 25.1976
Quantile 0.75 25.3181
Quantile 0.975 25.7377
Mean 24.4064
Stdev 0.240272
Quantile 0.025 23.8944
Quantile 0.25 24.2865
Quantile 0.5 24.4024
Quantile 0.75 24.5203
Quantile 0.975 24.9421
Mean 35.2087
Stdev 0.346389
Quantile 0.025 34.47
Quantile 0.25 35.0359
Quantile 0.5 35.2032
Quantile 0.75 35.3732
Quantile 0.975 35.9804
Mean 32.3883
Stdev 0.318249
Quantile 0.025 31.6798
Quantile 0.25 32.2348
Quantile 0.5 32.3914
Quantile 0.75 32.5452
Quantile 0.975 33.0673
Mean 32.002
Stdev 0.315016
Quantile 0.025 31.321
Quantile 0.25 31.8465
Quantile 0.5 31.9994
Quantile 0.75 32.1532
Quantile 0.975 32.6949
Mean 33.2137
Stdev 0.327071
Quantile 0.025 32.5246
Quantile 0.25 33.0491
Quantile 0.5 33.2063
Quantile 0.75 33.3674
Quantile 0.975 33.9504
Mean 33.1039
Stdev 0.325306
Quantile 0.025 32.4035
Quantile 0.25 32.9427
Quantile 0.5 33.1005
Quantile 0.75 33.2597
Quantile 0.975 33.822
Mean 29.1059
Stdev 0.286112
Quantile 0.025 28.4938
Quantile 0.25 28.9635
Quantile 0.5 29.1019
Quantile 0.75 29.2423
Quantile 0.975 29.7413
Mean 35.0898
Stdev 0.345157
Quantile 0.025 34.325
Quantile 0.25 34.9227
Quantile 0.5 35.0921
Quantile 0.75 35.2592
Quantile 0.975 35.8302
Mean 45.3864
Stdev 0.445648
Quantile 0.025 44.3983
Quantile 0.25 45.1705
Quantile 0.5 45.3896
Quantile 0.75 45.6054
Quantile 0.975 46.3413
Mean 35.3827
Stdev 0.348052
Quantile 0.025 34.6008
Quantile 0.25 35.216
Quantile 0.5 35.3879
Quantile 0.75 35.5555
Quantile 0.975 36.1181
Mean 45.9676
Stdev 0.452762
Quantile 0.025 44.9362
Quantile 0.25 45.7536
Quantile 0.5 45.9785
Quantile 0.75 46.1951
Quantile 0.975 46.9091
Mean 49.9462
Stdev 0.492939
Quantile 0.025 48.7972
Quantile 0.25 49.7182
Quantile 0.5 49.9657
Quantile 0.75 50.1992
Quantile 0.975 50.9421
Mean 32.1974
Stdev 0.316662
Quantile 0.025 31.5057
Quantile 0.25 32.0423
Quantile 0.5 32.1967
Quantile 0.75 32.3508
Quantile 0.975 32.8868
Mean 22.0095
Stdev 0.217194
Quantile 0.025 21.5526
Quantile 0.25 21.9002
Quantile 0.5 22.0044
Quantile 0.75 22.1113
Quantile 0.975 22.4995
Mean 20.1085
Stdev 0.198288
Quantile 0.025 19.6911
Quantile 0.25 20.0087
Quantile 0.5 20.1039
Quantile 0.75 20.2016
Quantile 0.975 20.5555
Mean 23.197
Stdev 0.228287
Quantile 0.025 22.6967
Quantile 0.25 23.0855
Quantile 0.5 23.197
Quantile 0.75 23.3079
Quantile 0.975 23.6925
Mean 22.3053
Stdev 0.219564
Quantile 0.025 21.8367
Quantile 0.25 22.1959
Quantile 0.5 22.3019
Quantile 0.75 22.4096
Quantile 0.975 22.7941
Mean 24.8084
Stdev 0.244035
Quantile 0.025 24.2914
Quantile 0.25 24.686
Quantile 0.5 24.8036
Quantile 0.75 24.9237
Quantile 0.975 25.3553
Mean 28.5093
Stdev 0.280418
Quantile 0.025 27.9147
Quantile 0.25 28.3687
Quantile 0.5 28.5039
Quantile 0.75 28.6419
Quantile 0.975 29.1371
Mean 37.2871
Stdev 0.366818
Quantile 0.025 36.4711
Quantile 0.25 37.1101
Quantile 0.5 37.2904
Quantile 0.75 37.4676
Quantile 0.975 38.0706
Mean 27.8967
Stdev 0.27371
Quantile 0.025 27.2975
Quantile 0.25 27.7627
Quantile 0.5 27.8966
Quantile 0.75 28.0298
Quantile 0.975 28.4909
Mean 23.8965
Stdev 0.234731
Quantile 0.025 23.3816
Quantile 0.25 23.7819
Quantile 0.5 23.8967
Quantile 0.75 24.0108
Quantile 0.975 24.4052
Mean 21.7064
Stdev 0.213731
Quantile 0.025 21.2521
Quantile 0.25 21.5995
Quantile 0.5 21.7026
Quantile 0.75 21.8075
Quantile 0.975 22.184
Mean 28.5983
Stdev 0.281643
Quantile 0.025 27.9841
Quantile 0.25 28.4603
Quantile 0.5 28.5975
Quantile 0.75 28.7345
Quantile 0.975 29.2127
Mean 27.0947
Stdev 0.266152
Quantile 0.025 26.5089
Quantile 0.25 26.9651
Quantile 0.5 27.0954
Quantile 0.75 27.2247
Quantile 0.975 27.6694
Mean 20.3021
Stdev 0.19987
Quantile 0.025 19.8712
Quantile 0.25 20.2032
Quantile 0.5 20.3001
Quantile 0.75 20.3978
Quantile 0.975 20.743
Mean 22.5038
Stdev 0.221179
Quantile 0.025 22.0294
Quantile 0.25 22.3939
Quantile 0.5 22.501
Quantile 0.75 22.6094
Quantile 0.975 22.9938
Mean 28.998
Stdev 0.285779
Quantile 0.025 28.3743
Quantile 0.25 28.8581
Quantile 0.5 28.9973
Quantile 0.75 29.1362
Quantile 0.975 29.6209
Mean 24.8034
Stdev 0.244378
Quantile 0.025 24.278
Quantile 0.25 24.6824
Quantile 0.5 24.8007
Quantile 0.75 24.9201
Quantile 0.975 25.3439
Mean 22.0156
Stdev 0.217377
Quantile 0.025 21.568
Quantile 0.25 21.9046
Quantile 0.5 22.008
Quantile 0.75 22.1158
Quantile 0.975 22.5148
Mean 26.3945
Stdev 0.259763
Quantile 0.025 25.822
Quantile 0.25 26.2682
Quantile 0.5 26.3953
Quantile 0.75 26.5212
Quantile 0.975 26.955
Mean 33.0796
Stdev 0.325243
Quantile 0.025 32.3431
Quantile 0.25 32.925
Quantile 0.5 33.0862
Quantile 0.75 33.2424
Quantile 0.975 33.7604
Mean 36.0749
Stdev 0.354885
Quantile 0.025 35.2671
Quantile 0.25 35.907
Quantile 0.5 36.0833
Quantile 0.75 36.2533
Quantile 0.975 36.8133
Mean 28.3907
Stdev 0.279111
Quantile 0.025 27.7707
Quantile 0.25 28.2558
Quantile 0.5 28.393
Quantile 0.75 28.528
Quantile 0.975 28.9877
Mean 33.3961
Stdev 0.328249
Quantile 0.025 32.6774
Quantile 0.25 33.2356
Quantile 0.5 33.3959
Quantile 0.75 33.5556
Quantile 0.975 34.1089
Mean 28.1907
Stdev 0.276822
Quantile 0.025 27.5757
Quantile 0.25 28.0568
Quantile 0.5 28.193
Quantile 0.75 28.327
Quantile 0.975 28.7825
Mean 22.8072
Stdev 0.224457
Quantile 0.025 22.331
Quantile 0.25 22.6948
Quantile 0.5 22.803
Quantile 0.75 22.9134
Quantile 0.975 23.3096
Mean 20.2974
Stdev 0.199378
Quantile 0.025 19.8607
Quantile 0.25 20.1999
Quantile 0.5 20.2974
Quantile 0.75 20.3944
Quantile 0.975 20.7301
Mean 16.0907
Stdev 0.15836
Quantile 0.025 15.733
Quantile 0.25 16.0153
Quantile 0.5 16.0937
Quantile 0.75 16.1697
Quantile 0.975 16.4232
Mean 22.0983
Stdev 0.217627
Quantile 0.025 21.6231
Quantile 0.25 21.9918
Quantile 0.5 22.0978
Quantile 0.75 22.2036
Quantile 0.975 22.5724
Mean 19.4047
Stdev 0.191138
Quantile 0.025 18.9968
Quantile 0.25 19.3094
Quantile 0.5 19.4017
Quantile 0.75 19.4954
Quantile 0.975 19.8303
Mean 21.6
Stdev 0.212196
Quantile 0.025 21.1393
Quantile 0.25 21.4955
Quantile 0.5 21.5988
Quantile 0.75 21.7024
Quantile 0.975 22.0646
Mean 23.7967
Stdev 0.233891
Quantile 0.025 23.2837
Quantile 0.25 23.6824
Quantile 0.5 23.7968
Quantile 0.75 23.9104
Quantile 0.975 24.3037
Mean 16.2052
Stdev 0.159694
Quantile 0.025 15.8665
Quantile 0.25 16.1253
Quantile 0.5 16.2022
Quantile 0.75 16.2807
Quantile 0.975 16.5629
Mean 17.8053
Stdev 0.17521
Quantile 0.025 17.433
Quantile 0.25 17.7176
Quantile 0.5 17.8021
Quantile 0.75 17.8882
Quantile 0.975 18.1969
Mean 19.7981
Stdev 0.194363
Quantile 0.025 19.3733
Quantile 0.25 19.7029
Quantile 0.5 19.7978
Quantile 0.75 19.8925
Quantile 0.975 20.2208
Mean 23.0989
Stdev 0.227184
Quantile 0.025 22.604
Quantile 0.25 22.9874
Quantile 0.5 23.0981
Quantile 0.75 23.2088
Quantile 0.975 23.5948
Mean 21.0062
Stdev 0.206346
Quantile 0.025 20.5677
Quantile 0.25 20.9029
Quantile 0.5 21.0025
Quantile 0.75 21.104
Quantile 0.975 21.4672
Mean 23.8004
Stdev 0.23426
Quantile 0.025 23.2924
Quantile 0.25 23.6851
Quantile 0.5 23.799
Quantile 0.75 23.9132
Quantile 0.975 24.3141
Mean 23.1023
Stdev 0.227285
Quantile 0.025 22.6122
Quantile 0.25 22.9898
Quantile 0.5 23.1001
Quantile 0.75 23.2112
Quantile 0.975 23.6034
Mean 20.4031
Stdev 0.200898
Quantile 0.025 19.9717
Quantile 0.25 20.3035
Quantile 0.5 20.4007
Quantile 0.75 20.499
Quantile 0.975 20.8478
Mean 18.5017
Stdev 0.181798
Quantile 0.025 18.1096
Quantile 0.25 18.4117
Quantile 0.5 18.5
Quantile 0.75 18.5889
Quantile 0.975 18.9023
Mean 24.9919
Stdev 0.245406
Quantile 0.025 24.447
Quantile 0.25 24.8732
Quantile 0.5 24.9939
Quantile 0.75 25.1127
Quantile 0.975 25.5169
Mean 24.5937
Stdev 0.241603
Quantile 0.025 24.0595
Quantile 0.25 24.4764
Quantile 0.5 24.595
Quantile 0.75 24.7121
Quantile 0.975 25.113
Mean 23.0003
Stdev 0.226234
Quantile 0.025 22.5094
Quantile 0.25 22.8889
Quantile 0.5 22.9989
Quantile 0.75 23.1093
Quantile 0.975 23.4961
Mean 22.1984
Stdev 0.218187
Quantile 0.025 21.7223
Quantile 0.25 22.0914
Quantile 0.5 22.1979
Quantile 0.75 22.3041
Quantile 0.975 22.6738
Mean 19.3033
Stdev 0.189958
Quantile 0.025 18.8959
Quantile 0.25 19.2089
Quantile 0.5 19.3009
Quantile 0.75 19.3939
Quantile 0.975 19.7243
Mean 22.6068
Stdev 0.222465
Quantile 0.025 22.1343
Quantile 0.25 22.4955
Quantile 0.5 22.6028
Quantile 0.75 22.7121
Quantile 0.975 23.1042
Mean 19.8078
Stdev 0.194915
Quantile 0.025 19.3966
Quantile 0.25 19.7098
Quantile 0.5 19.8035
Quantile 0.75 19.8996
Quantile 0.975 20.2463
Mean 17.1019
Stdev 0.168604
Quantile 0.025 16.7388
Quantile 0.25 17.0186
Quantile 0.5 17.1002
Quantile 0.75 17.1826
Quantile 0.975 17.4742
Mean 19.4029
Stdev 0.191047
Quantile 0.025 18.9925
Quantile 0.25 19.3082
Quantile 0.5 19.4007
Quantile 0.75 19.4941
Quantile 0.975 19.8257
Mean 22.2007
Stdev 0.21852
Quantile 0.025 21.7272
Quantile 0.25 22.093
Quantile 0.5 22.1992
Quantile 0.75 22.3058
Quantile 0.975 22.6803
Mean 20.7055
Stdev 0.203974
Quantile 0.025 20.2709
Quantile 0.25 20.6037
Quantile 0.5 20.7021
Quantile 0.75 20.8022
Quantile 0.975 21.1604
Mean 21.1005
Stdev 0.207425
Quantile 0.025 20.651
Quantile 0.25 20.9983
Quantile 0.5 21.0992
Quantile 0.75 21.2004
Quantile 0.975 21.5556
Mean 19.503
Stdev 0.19179
Quantile 0.025 19.0911
Quantile 0.25 19.4078
Quantile 0.5 19.5007
Quantile 0.75 19.5946
Quantile 0.975 19.9274
Mean 18.5065
Stdev 0.182403
Quantile 0.025 18.1205
Quantile 0.25 18.4151
Quantile 0.5 18.5028
Quantile 0.75 18.5925
Quantile 0.975 18.9158
Mean 20.6025
Stdev 0.202458
Quantile 0.025 20.1668
Quantile 0.25 20.5021
Quantile 0.5 20.6004
Quantile 0.75 20.6995
Quantile 0.975 21.0496
Mean 19.0022
Stdev 0.18696
Quantile 0.025 18.5997
Quantile 0.25 18.9097
Quantile 0.5 19.0003
Quantile 0.75 19.0917
Quantile 0.975 19.4151
Mean 18.6977
Stdev 0.183889
Quantile 0.025 18.295
Quantile 0.25 18.6078
Quantile 0.5 18.6977
Quantile 0.75 18.787
Quantile 0.975 19.097
Mean 32.6854
Stdev 0.321868
Quantile 0.025 31.9645
Quantile 0.25 32.5311
Quantile 0.5 32.6897
Quantile 0.75 32.8447
Quantile 0.975 33.3679
Mean 16.5129
Stdev 0.163492
Quantile 0.025 16.1779
Quantile 0.25 16.4292
Quantile 0.5 16.5067
Quantile 0.75 16.5878
Quantile 0.975 16.8901
Mean 23.9005
Stdev 0.234947
Quantile 0.025 23.3912
Quantile 0.25 23.7847
Quantile 0.5 23.899
Quantile 0.75 24.0137
Quantile 0.975 24.4158
Mean 31.1673
Stdev 0.307493
Quantile 0.025 30.4517
Quantile 0.25 31.0248
Quantile 0.5 31.1791
Quantile 0.75 31.3248
Quantile 0.975 31.7898
Mean 17.503
Stdev 0.172591
Quantile 0.025 17.1327
Quantile 0.25 17.4174
Quantile 0.5 17.5008
Quantile 0.75 17.5852
Quantile 0.975 17.8855
Mean 17.2083
Stdev 0.169656
Quantile 0.025 16.8528
Quantile 0.25 17.1227
Quantile 0.5 17.204
Quantile 0.75 17.2877
Quantile 0.975 17.5923
Mean 23.0975
Stdev 0.227309
Quantile 0.025 22.6001
Quantile 0.25 22.9864
Quantile 0.5 23.0973
Quantile 0.75 23.2078
Quantile 0.975 23.5915
Mean 24.4999
Stdev 0.240682
Quantile 0.025 23.9771
Quantile 0.25 24.3814
Quantile 0.5 24.4986
Quantile 0.75 24.616
Quantile 0.975 25.0267
Mean 26.6052
Stdev 0.262009
Quantile 0.025 26.0441
Quantile 0.25 26.4749
Quantile 0.5 26.6016
Quantile 0.75 26.7299
Quantile 0.975 27.1868
Mean 22.902
Stdev 0.225454
Quantile 0.025 22.4155
Quantile 0.25 22.7906
Quantile 0.5 22.8999
Quantile 0.75 23.0101
Quantile 0.975 23.3988
Mean 24.1018
Stdev 0.237104
Quantile 0.025 23.5897
Quantile 0.25 23.9846
Quantile 0.5 24.0998
Quantile 0.75 24.2156
Quantile 0.975 24.6237
Mean 18.6057
Stdev 0.183453
Quantile 0.025 18.216
Quantile 0.25 18.514
Quantile 0.5 18.6023
Quantile 0.75 18.6924
Quantile 0.975 19.016
Mean 30.0853
Stdev 0.296318
Quantile 0.025 29.4198
Quantile 0.25 29.9436
Quantile 0.5 30.0898
Quantile 0.75 30.2323
Quantile 0.975 30.7118
Mean 18.2052
Stdev 0.179249
Quantile 0.025 17.8239
Quantile 0.25 18.1156
Quantile 0.5 18.2021
Quantile 0.75 18.2901
Quantile 0.975 18.6055
Mean 20.6026
Stdev 0.203125
Quantile 0.025 20.1655
Quantile 0.25 20.5021
Quantile 0.5 20.6004
Quantile 0.75 20.6997
Quantile 0.975 21.0516 # Posterior mean and 95% credible interval
map$PMoriginal <- sapply(marginals_summaries, '[[', "mean")
map$LLoriginal <- sapply(marginals_summaries, '[[', "quant0.025")
map$ULoriginal <- sapply(marginals_summaries, '[[', "quant0.975")# Common legend
at <- seq(
min(c(map$PMoriginal, map$LLoriginal, map$ULoriginal)),
max(c(map$PMoriginal, map$LLoriginal, map$ULoriginal)),
length.out = 8
)
# Popup table
popuptable <- leafpop::popupTable(
dplyr::mutate_if(map, is.numeric, round, digits = 2),
zcol = c("TOWN", "vble", "CRIM", "RM", "PM", "LL", "UL"),
row.numbers = FALSE,
feature.id = FALSE
)
# Map visualizations
m1 <- mapview(
map,
zcol = "PMoriginal",
map.types = "CartoDB.Positron",
at = at,
popup = popuptable
)
m2 <- mapview(
map,
zcol = "LLoriginal",
map.types = "CartoDB.Positron",
at = at,
popup = popuptable
)
m3 <- mapview(
map,
zcol = "ULoriginal",
map.types = "CartoDB.Positron",
at = at,
popup = popuptable
)m <- leafsync::sync(m1, m2, m3, ncol = 3)
mPosterior mean of the housing prices (left), together with lower (center) and upper (right) limits of 95% credible intervals.
Spatial disease risk models are commonly specified using a Poisson distribution for the observed number of cases (Yi) with mean equal to the expected number of cases (Ei) times the relative risk (??i) corresponding to area i, i = 1,…,n,
library(SpatialEpi)Warning: package 'SpatialEpi' was built under R version 4.4.3Loading required package: spdata(pennLC)
class(pennLC)[1] "list"names(pennLC)[1] "geo" "data" "smoking" "spatial.polygon"head(pennLC$data) county cases population race gender age
1 adams 0 1492 o f Under.40
2 adams 0 365 o f 40.59
3 adams 1 68 o f 60.69
4 adams 0 73 o f 70+
5 adams 0 23351 w f Under.40
6 adams 5 12136 w f 40.59head(pennLC$smoking) county smoking
1 adams 0.234
2 allegheny 0.245
3 armstrong 0.250
4 beaver 0.276
5 bedford 0.228
6 berks 0.249library(sf)
map <- st_as_sf(pennLC$spatial.polygon)
countynames <- sapply(
slot(pennLC$spatial.polygon, "polygons"),
function(x) { slot(x, "ID") }
)
map$county <- countynames
head(map)Simple feature collection with 6 features and 1 field
Geometry type: POLYGON
Dimension: XY
Bounding box: xmin: -80.51776 ymin: 39.72889 xmax: -75.53303 ymax: 41.1441
Geodetic CRS: +proj=longlat
geometry county
1 POLYGON ((-77.4467 39.96954... adams
2 POLYGON ((-80.14534 40.6742... allegheny
3 POLYGON ((-79.21142 40.9091... armstrong
4 POLYGON ((-80.1568 40.85189... beaver
5 POLYGON ((-78.38063 39.7288... bedford
6 POLYGON ((-75.53303 40.4508... berksObserved cases
library(dplyr)
Attaching package: 'dplyr'The following objects are masked from 'package:stats':
filter, lagThe following objects are masked from 'package:base':
intersect, setdiff, setequal, uniond <- group_by(pennLC$data, county) %>% summarize(Y= sum(cases))
head(d)# A tibble: 6 × 2
county Y
<fct> <int>
1 adams 55
2 allegheny 1275
3 armstrong 49
4 beaver 172
5 bedford 37
6 berks 308Expected cases
pennLC$data <- pennLC$data[order(pennLC$data$county,
pennLC$data$race, pennLC$data$gender, pennLC$data$age), ]E <- expected(population = pennLC$data$population,
cases = pennLC$data$cases, n.strata = 16)
d$E <- E
head(d)# A tibble: 6 × 3
county Y E
<fct> <int> <dbl>
1 adams 55 69.6
2 allegheny 1275 1182.
3 armstrong 49 67.6
4 beaver 172 173.
5 bedford 37 44.2
6 berks 308 301. Smokers proportions
d <- dplyr::left_join(d, pennLC$smoking, by = "county")Standardized Mortality Ratios
d$SMR <- d$Y/d$E
head(d)# A tibble: 6 × 5
county Y E smoking SMR
<fct> <int> <dbl> <dbl> <dbl>
1 adams 55 69.6 0.234 0.790
2 allegheny 1275 1182. 0.245 1.08
3 armstrong 49 67.6 0.25 0.725
4 beaver 172 173. 0.276 0.997
5 bedford 37 44.2 0.228 0.837
6 berks 308 301. 0.249 1.02 map <- dplyr::left_join(map, d, by = "county")library(mapview)
library(RColorBrewer)
pal <- colorRampPalette(brewer.pal(9, "YlOrRd"))
mapview(
map,
zcol = "SMR",
color = "gray",
alpha.regions = 0.8,
layer.name = "SMR",
col.regions = pal,
map.types = "CartoDB.Positron"
)Relative risks of the counties of Pennsylvania, USA.
library(mapview)
library(RColorBrewer)
library(leafpop)
pal <- colorRampPalette(brewer.pal(9, "YlOrRd"))
mapviewOptions(fgb = FALSE)
popuptable <- leafpop::popupTable(
dplyr::mutate_if(map, is.numeric, round, digits = 2),
zcol = c("county", "Y", "E", "smoking", "SMR"),
row.numbers = FALSE,
feature.id = FALSE
)
mapview(
map,
zcol = "SMR",
color = "gray",
col.regions = pal,
highlight = leaflet::highlightOptions(weight = 4),
popup = popuptable
)library(spdep)
library(INLA)
nb <- poly2nb(map)
nb2INLA("map.adj", nb)
g <- inla.read.graph(filename = "map.adj")map$re_u <- 1:nrow(map)
map$re_v <- 1:nrow(map)formula <- Y ~ smoking + f(re_u, model = "besag", graph = g, scale.model = TRUE) + f(re_v, model = "iid")res <- inla(formula, family = "poisson", data = map, E= E, control.predictor = list(compute = TRUE), control.compute = list(return.marginals.predictor = TRUE))res$summary.fixed mean sd 0.025quant 0.5quant 0.975quant mode
(Intercept) -0.3235144 0.1499851 -0.61968472 -0.3233618 -0.02842592 -0.3234193
smoking 1.1545904 0.6234222 -0.07736443 1.1559465 2.38003002 1.1562136
kld
(Intercept) 3.572898e-08
smoking 3.580630e-08We see the intercept ??0 =-0.323 with a 95% credible interval equal to (-0.619,-0.029), and the coefficient of smoking is ??1 = 1.155 with a 95% credible interval equal to (-0.076, 2.378) This indicates a non-significant effect of smoking.
res$summary.fitted.values[1:3,] mean sd 0.025quant 0.5quant 0.975quant
fitted.Predictor.01 0.8779225 0.05818241 0.7644309 0.8776429 0.9936634
fitted.Predictor.02 1.0597610 0.02751494 1.0072677 1.0592514 1.1151485
fitted.Predictor.03 0.9644514 0.05103596 0.8598242 0.9656337 1.0622242
mode
fitted.Predictor.01 0.8776984
fitted.Predictor.02 1.0582241
fitted.Predictor.03 0.9680030# Relative risk
map$RR <- res$summary.fitted.values[, "mean"]
# Lower and upper limits of 95% credible intervals
map$LL <- res$summary.fitted.values[, "0.025quant"]
map$UL <- res$summary.fitted.values[, "0.975quant"]library(mapview)
library(RColorBrewer)
library(leafpop)
pal <- colorRampPalette(brewer.pal(9, "YlOrRd"))
mapviewOptions(fgb = FALSE)
mapview(
map,
zcol = "RR",
color = "gray",
col.regions = pal,
highlight = leaflet::highlightOptions(weight = 4),
popup = leafpop::popupTable(
dplyr::mutate_if(map, is.numeric, round, digits = 2),
zcol = c("county", "Y", "E", "smoking", "SMR", "RR", "LL", "UL"),
row.numbers = FALSE,
feature.id = FALSE
)
)Relative risks of the counties of Pennsylvania, USA.
Comparing SMR and RR maps
at <- seq(min(map$SMR), max(map$SMR), length.out = 8)
m1 <- mapview(
map,
zcol = "SMR",
color = "gray",
col.regions = pal,
at = at
)
m2 <- mapview(
map,
zcol = "RR",
color = "gray",
col.regions = pal,
at = at
)
leafsync::sync(m1, m2)SMRs (left) and RRs (right) of the counties of Pennsylvania, USA.
Exceedance probabilities
c<-1.2
marg<-res$marginals.fitted.values[[51]]
1-inla.pmarginal(q = c,marginal= marg)[1] 0.05636408library(ggplot2)
marginal <- inla.smarginal(res$marginals.fitted.values[[51]])
marginal <- data.frame(marginal)
ggplot(marginal, aes(x = x, y = y)) +
geom_line() +
labs(x = expression(theta[51]), y = "Density") +
geom_vline(xintercept = 1.2, col = "black") +
theme_bw(base_size = 20)
Posterior distribution of the relative risk for area 51 exceeding the threshold value of 1.2. The vertical line indicates the threshold value.
c<-1.2
map$exc<-sapply(res$marginals.fitted.values,
FUN= function(marg){1-inla.pmarginal(q=c,marginal= marg)})pal <- colorRampPalette(brewer.pal(9, "YlOrRd"))
mapview(map, zcol = "exc", color = "gray", col.regions = pal, map.types = "CartoDB.Positron")Probabilities that the relative risks of counties exceed 1.2.
Spatial analyses of aggregated data often face the Misaligned Data Problem (MIDP), which occurs when the scale of the analyzed data differs from the scale at which it was originally collected. This misalignment can result in a loss of spatial detail, potentially obscuring important patterns and leading to biased or misleading conclusions (Banerjee et al., 2004).
Another common issue is the Modifiable Areal Unit Problem (MAUP) (Openshaw, 1984), where results vary depending on the level or configuration of spatial aggregation. The MAUP consists of two effects: the scale effect, where results change with the level of aggregation, and the zoning effect, where arbitrary boundary definitions influence outcomes.
Ecological studies (Robinson, 1950) analyze relationships between exposures and outcomes at the group level rather than the individual level. While useful when individual data are unavailable, they are prone to the ecological fallacy-group-level associations that do not necessarily apply to individuals. This introduces ecological bias, a specific form of MAUP that includes aggregation and specification biases (Gotway & Young, 2002).
Finally, integrating spatial data from different sources or resolutions-such as monitoring stations and satellite imagery-can improve prediction accuracy. Moraga et al. (2017) proposed a Bayesian melding model combining spatially misaligned data using INLA and SPDE for efficient inference. Zhong and Moraga (2023) compared this model with a Bayesian downscaler, demonstrating its ability to disaggregate areal data and produce spatially continuous predictions that enhance policy-relevant decision-making.